How do I get my URL to show up in search engines?

Did you follow the rest of the instructions https://support.google.com/webmasters/answer/183669?hl=en and https://support.google.com/webmasters/answer/34444?hl=en#1

Did you check in webmaster tools if there are any issues?

Is this a brand new domain?  

What  happens if you google "mydomain.com" with the quotes?

It does take some time, but 2 to 3 weeks sounds about right.

Thanks - I guess I need to be a little more patient.

I have not checked in the webmaster tools.  Do I need to use the webmaster tools to submit a Sitemap.xml or can I do it just by pinging the various search engines?

No, this is not a brand new domain, has been around for about 6 months.

If I google the URL using quotes it does come up.

>When I enter my url in Google's my website is not showing up in the list.

I think something is lost in translation of this sentence. I thought your domain was not coming up at all.  If you google your domain and the site comes up, you are in google's index.  

What is your domain name?

What are you entering into google or bing as a search phrase?

Did you have a site map in the past?  If you did, you need to resubmit a site map if your site has changed.

Do you have a robots.txt?  If so, are you blocking anything?

Did you go through this link https://support.google.com/webmasters/answer/34444?hl=en#1

This is actually not my site, I am trying to help a friend.

The site is art.mojo.ca.  

If I enter just art.mojo.ca in the search bar the site doesn't show up in the search results but lots of others do instead.

If I enter "art.mojo.ca" in the search bar then it shows up in the search results

The site did not have a sitmap in the past, I was submitting it for my friend for the first time.

Robots.txt just contains the following:
User-agent: *
Crawl-delay: 10

I have not gone through the link you provided, I will do that

There is almost nothing there to index, and with the HTML validator reporting 342 errors, it is unlikely that the spider could successfully crawl the page anyway.

You can follow the steps that padas has given you to get the sitemap on Google but you are not going to be very successful having it show up in any search with the code in that condition.

Cd&

The main issue is the domain is mojo.ca which is redirected to http://www.adreflex.com.  If you don't own this domain, there is not much you can do.  Forget about the fact there is no content on the home page.

If all you want to do is to place your domain name to google and have it show up, you will need to get your own domain name and see if adreflex will allow you to use your domain name.

I was not very impressed with the adreflex site. Check out http://www.weebly.com/, http://wix.com or http://www.squarespace.com.  They all offer an easy way to make your own site and I think they do it in a much better way.  All will allow you to use your own domain name as well.

One thing you will want to do is convert your long urls to friendly urls so http://www.art.mojo.ca/index.cfm?page_id=24&Label=About%20Art converts to http://www.art.mojo.ca/about_art/

and
http://www.art.mojo.ca/index.cfm?FuseAction=DspPhotos&prodgroup=cf_photo&hidgalleryid=5&CFID=30170709&CFTOKEN=63691a7f20886dae-E6111B8F-E46B-1DEC-0E6B1FD5D260700D&jsessionid=843095cbe246c6fb8e166b2b26124df91e53
converts to
http://www.art.mojo.ca/

If they can't do that for you, find a different provider.

To see if a brand new site is indexed by Google, open a Webmaster Tools Account, add your site and site map. Their interface will tell you how many of your pages are indexed in Google from your site map.

However, having your site indexed does not guarantee it will "show up" in search engine result pages with any decent ranking. In general, to show well, you need inbound links from other high quality sites, great content and a good user experience.

Webmaster tools will not help in this case since he does not own the domain.

To share artwork, deviantart.com is your best bet.

Sample site you can create with this tool - http://adamhughes.deviantart.com/

Do a site:adamhughes.deviantart.com on Google and you'll see 500+ pages indexed, pretty much addresses your friend's indexing issue :)

Changing the home page solved the problem