Hi Derekk:
Unfortunately your question is too broad to give a simple answer and I don't know the exact variables and what you need but in general:
If you are only to have a simple free, nice, safe gallery, for your own personal use, you may start with a site such as here (WorldPhotographyForum - Gallery), Flickr (Flickr is sold to Verizon and things may change soon) or Pixieset .
If you are familiar with web stuff, you may want to try
jAlbum.net for free.
If you need more space and willing to pay, look into a paid
Image Hosting Site such as
Photobucket, or
try
Smugmug or likewise.
If you are more expert and can manage codes etc, you may try
BlueHost currently starting at $3 per month.
You probably need to own your domain so look into something like
GoDaddy or likewise.
And finally, if you are really too stranger to web stuff, you may find hosting your own site might be way overrated!!! In that case, look into your local advertisements to see if you can find a professional IT that can assist and create the site for you, for cheap.
Best of luck