First I would suggest always going with a hosting service, because their in the business of keeping the server up and running. Whereas if you host at home one trip of the plug or power outage will bring your site down. Also this way you always have a backup somewhere else, if say your computer gets stolen or crashes.
As for hosting services, there are a lot of options. I recently came across this article on hosting services that you may find useful http://www.consumer-rankings.com/hosting/
Personally I use Media Temple (mediatemple.net) which is also what I’ve had the department purchase for our web courses. Media Temple sells servers space as opposed to hosting. Most hosting space is for one site, while server space allows you to host multiple site. On my personal account I use the smallest package which allows me to host up to 100 websites, while our school account allows for more than that and is set up as a hosting resell (meaning you sell hosting space on your server).
If you plan on making money on building websites then servers space is the way to go. For example my personal account monthly cost is $20 a month, thus if I can at least get 4 client websites hosted at $5 a month each, then I’ll break even, if I have more clients I get a profit (each month). Really up to you, $20 a month might be too much if you just want to play around since there are other sites that offer domain and hosting at a much smaller monthly fee.
One thing I do encourage you to look into is whether or not the service you choose support Content Management Systems (CMS), although you might start with a basic HTML site, if you plan on expanding your site CMS is the way to go. WordPress is the most popular CMS out there currently and many hosting sites now offer a one click install of the system. This makes setting up the site even more convenient.
These are just my personal thoughts on the matter, though I do encourage you to research all options and even post this question to other web designers, and on web message forums for more insight on the topic.