I do not consider myself a pro, especially since I usually depend on the work of others, but here is what I recently told a friend who was interested in putting some pictures on a website.  I would also appreciate any comments on better ways to do this, and feel free to click the referrer links when you set up your own accounts, as I get a little from that, and in the case of smugmug you get a cheaper price too.
 I am currently using Gallery2 (http://gallery.menalto.com/) to show off my pictures.  I have my pictures and the Gallery hosted on a webhosting account with Dreamhost (http://www.dreamhost.com/r.cgi?80130).  I have modified some things from a default installation, but Dreamhost has an easy way to install Gallery2 on your account.  Dreamhost charges monthly or you can pay for one or two years up front.  The cheapest plan is $9.95/month or $7.95/month if you prepay for two years.  I have been using Dreamhost for a little over a year to host my website, and it has been a great experience.  If you do use Dreamhost then please put me down as your referrer by using this link: http://www.dreamhost.com/r.cgi?80130
I have used smugmug to host my pictures, and they are actually still mostly there too (http://carruth.smugmug.com).  I have also had a good experience there, and it is much better and easier if you just want to put pictures up somewhere.  It is $39.95/year for the cheapest plan (which I am on).  You can upload unlimited photos and they limit the bandwidth that people looking at your photos can use (never been a problem for me).  If you use smugmug then please put me down as your referrer by using this link to signup (http://www.smugmug.com/?referrer=vIVgjqIX0yvV).  If you put me down as your referrer for this, then you save $5 off the annual fee.  Also, smugmug provides a free 7 day trial.
My recommendation depends on what you are looking for.  They both have a pretty good layout and system, but smugmug is setup to be very easy to use and not require any knowledge of web design to use.  If you only want to put pictures, then I would recommend smugmug.  It is a great service.  If you want to host a website with photos and other pages and stuff, then you may want to use a webhost.  Dreamhost may not be the cheapest out there, but I use it for lots of stuff and they give me plenty of space (more than 20 GB) for all of my pictures.  I also use dreamhost to host several other websites (www.eldercarruth.com, www.sistercarruth.com, www.woodspringforest.com, www.edehaiti.org), and I need to be able to do more than just post pictures.  Dreamhost is the only webhost I have used, but I have been very impressed.
Another option that you may want to consider.  Many times, whichever company you use for internet access (i.e. RoadRunner, DSL, or even dialup) will give you a small amount of space to put a website up.  That is usually not enough space for pictures, but you could host your website on that space (websites are usually pretty small – mostly just text), and then get a smugmug account for your pictures.  I considered doing that, but now I host too many websites and it would be difficult to do that. 
Feel free to ask any other questions that come up, and let me know what you end up with.  I have enjoyed playing around with different things to get my website the way it is.  Good Luck.