Which SSD drive and which RAM modules should I buy?

Stick with one of the "names" ... Intel, Hitachi, Samsung, Seagate, Western Digital, etc.  There are any number of "brand X" or house brand drives on the market.  Some of those are ... "not good", e.g. the Hyundai Sapphire series with no cache and lacking the SSD SMART variables that tell when it's going to fail.

I admit that the day of the five-year drive warranty is over, but it is not prudent to put anything on an SSD with a 90-day or one-year warranty.

My advice is:  Before buying, go read independent reviews written by people who don't sell the drive.  And never rely on what you see on amazon.

I doubt there is much difference. The description for the SanDisk I've seen though emphasises that it is particularly fit for use in Servers & RAID arrays, while the Crucial Disk has no such information in it's description.

The speed is dependent on the Interface in your PC/Server/Laptop. If your System has SATA-III, then the two models already are at the top of the speeds you can expect, so for that Interface it is practically as fast as it can go.

If you want "Faster", then you'd need an M.2 connection on your mainboard that is NVME compatible, & get an NVME SSD.

First, I avoid buying from Best Buy.  The name suggests you're going to get a good deal (in theory, the "best"), but you can often find things cheaper elsewhere.  Additionally, there's been some suggestion that Best Buy sells products that are spec and functionally identical to similar models that are ONLY sold at Best Buy to ensure they don't ever have to match the price for from another vendor.  This doesn't always happen, but I've heard and am fairly certain I've seen it in the past. (I kid (somewhat) by explaining the reason I don't shop at Best Buy is that they lie to you before you even walk in the door).

Second, I agree with Dr. Klahn in terms of brands.  However, even the "top" brands have issues sometimes.  Western Digital recently had a bit of a scandal - see https://arstechnica.com/gadgets/2021/08/silent-changes-to-western-digitals-budget-ssd-may-lower-speeds-by-up-to-50/

Generally, Samsung has the best reputation for consumer grade stuff, though I've used Intel, Western Digital, San Disk and Kingston.

Where to buy?  If you have a Micro Center near you, I'd recommend them.  They have great prices on hard drives.  Otherwise, Amazon and NewEgg would generally be my preferences.
Here's a Samsung on NewEgg for $108 - https://www.newegg.com/p/0D9-0009-00AS7?Item=9SIAPBUEBY1775.
And at Micro Center, same drive, $115 - https://www.microcenter.com/product/624754/samsung-870-qvo-1tb-ssd-4-bit-mlc-v-nand-sata-iii-6gb-s-25-internal-solid-state-drive

Crucial are a well known brand, it is TLC with SLC buffer so performance is good except for sustained write, pretty much the same performance as the SanDisk one, It has power loss protection which is a bonus.

Often buying from a store versus ordering online is due to one question: when do I need it?
If the need is immidiate, I.e. Failing resource that needs to be replaced immidiately, I get it from the store.
If it can wait, order online.
Major brands ..
Often, each person may develop a preference based on experience.
One thing to take account is to make sure you have a backup setup.
Data recovery from failed SSD are extremely costly.

For me, going to a shop nearby is always better. You don't pay extra for delivery. besides that, you can pay cash, & normally you can also return it more easily if it doesn't work. Online you are always in doubt whether the seller really is serious, & whether he fully refunds things if necessary. You'd probably also have to pay extra for mail costs when you return something.

A nearby shop, if you have problems with it, you have a real address which you can report to authorities.

Lee W suggested something different than SATA.  
Here is a summary:
https://blogs.umass.edu/Techbytes/2018/02/23/types-of-ssds-and-which-ones-to-buy/
So, while SATA is sorta fast, others are even faster.

If the drive is not used for booting and you don't have a socket to support an NVMe device, you can get an inexpensive PCIe-to-NVMe adapter for under $20 US.  As Lee W suggested, the NVMe can be much faster than a SATA SSD.

There is a procedure to add NVMe boot capabilities to a system that doesn't already have it:
https://linustechtips.com/topic/592133-howto-get-nvme-support-on-older-motherboards/
This gets you around a common error:
https://www.win-raid.com/t1988f16-file-size-exceeds-the-volume-size-while-inserting-NvmeExpressDxE-compr.html

I've done this on a few older Dells with success.  I've run into one issue I've not gotten past, though.  Every boot gives a disk boot error and I have to hit F1 to continue.  Otherwise, it works fine.

The procedure is somewhat technical.  Read the instructions carefully and decide if you're up for it.

Crucial and SanDisk are decent "name" brands.  I've had both of them as well as the other major "name" brands and they all work just fine.  The NAND chips in the SSDS all come from basically the same factories.  The main difference is the packaging and the controller features.  Just don't buy from fly-by-night companies and you'll be fine.

This is the computer I am using

Model: HP Compaq Elite 8300 SFF
Processor: Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz
RAM: 8GB

Will it support anything faster than the Sandisk 3D SSD?

Not sure what you are asking, many ssd's sata Max around 540/500 M/s read/writes.

Your system can go to 32GB of ram.

https://support.hp.com/us-en/document/c03345460

Any SSD will outperform the HDD if one is in.
Sandisk was bought out by Western digital, the acronis imaging tool is available for download to clone.....

What does the system do, adding memory could further improve performance.

Rarely will you be processor bound...

From the Specs I've found on that PC, it looks as if it's SATA ports are SATA-II & not SATA-III. So if that is correct you won't get the full speed out of the SATA-III SSD's you have posted.

There are PCI Express NVMe cards available, & then you could use a fast NVMe SSD. But as has already been explained earlier, you might not be able to boot from it, or not easily:

https://www.newegg.com/riitop-model-ngftpce-bk-pci-express-controller-card/p/17Z-0061-00018?Description=pcie%20nvme%20adapter&cm_re=pcie_nvme%20adapter-_-9SIA6V85HC8902-_-Product

Rindi thank you. That looks like a good option.

What SSD should I buy with that adapter?

My thinking regarding getting it to boot would be to first update the BIOS on the machine and then set the boot order to check that drive first.

Would that be accurate or did I miss something else?

I think CompProbSolver has posted the Links you would need to get it to boot from the NVMe SSD. But why not boot from the SATA SSD you have already bought, & use the NVMe SSD as a 2nd Disk? Also, as arnold commented above, what is the PC used for, more RAM would probably give you the better speed boost...

You'd need an NVMe SSD with that adapter, but which one I'd get would depend on the capacity requirements. Something like this:

https://www.newegg.com/samsung-1tb-980/p/N82E16820147804?Description=nvme%20ssd&cm_re=nvme_ssd-_-20-147-804-_-Product

Thank you very much!  I'll review the instructions for getting it to boot and potentially buy that PCI Express adapter as well as the NVMe SSD from Samsung. That sounds like a good possibility.

Now I need to find the 32 GB I suppose and upgrade the memory. What would be your suggestion with respect to that? It looks like the board has four slots.

I'd go to the next PC shop in your area, and just ask for 4 x 8GB DDR3/1600 Modules. Maybe take one of your existing Modules with you so the shopkeeper has the dimensions.

https://www.newegg.com/p/pl?N=100007611%20600006050%20600551103%20601275375%2050001459%2050001455%2050001183&cm_sp=filter-_-INFOCARD-_-Subcategory-_-147-_-Property_600006050-_-4

Be careful if you order from the Link above, some of the offers there come in packages of 2 or 4, so that you don't order too many!

Ted,
You currently have 8GB, one thing to check is whether you have a single 8GB module installed, or a pair if 4GB.
Depending on the answer, if it is 2x4GB, get a pair of 8Gb, and get the system to 24GB

While 32GB is max, does [edited[ your use require that.

The Newegg link to the SAMSUNG 980 is misleading, it is not MLC it is TLC (Samsung cal it 3-bit MLC which is therefore TLC). It is DRAM-less but has a SLC buffer. You don't get the burst write performance without DRAM but you'll get about 2,500 MB/s until the SLC cache is full.

interesting comment. 2-bit seems to be the one that includes a Dram. was not paying attention to these...

Live and Learn new things

What are the important characteristics of RAM?
Which should we be buying?

So far I am understanding that I need to look for and confirm the following:

8GB modules
DDR3
1600
DRAM
SLC
MLC vs TLC

The one you need is based on the spec for your system; a DDR3-12800 (1600-MHz non-ECC DDR3)  you should look at the one you have dealing with timing, Cas .and module voltage.
1.35 or 1.5
HP does not specify which (likely either will work, though not clear whether you can mix modules.

Using the partnumber of the module you have could help identify the SPEC of what you have to then match the Timing, Cas and Voltage in the new ones if you are not buying all four modules...

You can find what the current RAM is without taking it apart, the SPD info from https://www.cpuid.com/softwares/cpu-z.html has the exact spec of your DIMM(s)

GREAT INFO.  THANK YOU!

I started this document also to try and help me sort it all out.
https://docs.google.com/document/d/1Juvn5H0ch0SQ08DJ2GLamCfWh0oh62lvZgxQN0kJnjM/edit

Still looking for the best solution for memory.

This is the memory report from the CPU-Z installation.

The SPD tab has more info, or you can hit Tools to get a text file.

If I am not misreading the info presented, this means you have a single 8GB module in the system.
meaning trying to buy 3x8 in a dual channel setup could pose an instability between the 8GB you already have and the 8GB you will get..... dual channel commonly means the two modules should be closely matched.....

You probably are better off buying the four,
https://www.newegg.com/g-skill-32gb-240-pin-ddr3-sdram/p/N82E16820231569

9-9-9 24 1.5V

RE PCI, and NVME, you need a low profile PCI-E for NVME
The SFF commonly has two PCI-E slots.

The following are options for NVMe.

https://www.newegg.com/p/pl?d=Low+profile+NVME+PCIE+card

To be bootable from NVME, on the PCI that is I think the issue Lee raised.

I am not sure whether the NVME on pci-E will be presented as a valid boot device unless you get the right pci-e

Searching for BOotable PCI-E NVME adapters, there are a few but are full size ...not low profile..

I think most PCIe cards just have a piece that makes them suitable for a small form factor and those devices come with both that are interchangeable.

If I get a regular size one I think I can make it into the smaller form factor.

I could be wrong about that.

I have updated the original question to show the current objective as well as what I have figured out so far.

Ted, usually, the PCI-E brackets are hard pressure-punched, only those that specify that they are interchangeable and both brackets Are included.

This is what I found out about the memory I have.

https://www.screencast.com/t/b93clmrM59ca.

What would I need based on that?

Suggested so far
4-8GB modules
DDR3
1600 ?
12800 ?
1600 MHZ
NON-ECC
DRAM
SLC
MLC vs TLC
TIMING: CAS AND MODULE VOLTAGE 1.35v or 1.5v (both might work but unclear)

Ok, you have 4x2GB PC3-10800 DDR3-1333.
4x8 is the way to go to get it to 32GB.

You could try selling the 4x2DDR3 after.

Can always buy a couple of 8GB ones and keep two 2GB for 20GB total.

SPD is serial presence detect, a low speed data line the chipset/CPU uses to interrogate the management ROM on the RAM. All those JEDEC data on that tab are the DIMM's capabilities, the CPU does not test each speed by writing to the RAM but asks the DIMM what it is capable of via the SPD bus. I said DIMM but they may be SODIMM, that info is only available in the CPU-Z text file, not the graphic version.