I'd second the recommendation of the CGI Resource Index, though I do think you'll probably find plenty there to be enthusiastic about. I love that site, and the PHP Resource Index too. You don't have to be a programmer to be able to install most of the scripts they list -- there are usually complete installation directions so that if you know how to use and FTP program and set permissions, you can pretty much get the job done.
If you're really knew and even that seems daunting, then yes, using a remotely hosted solution like the one Prufrock mentioned can be a reasonable alternative. You don't have as much control as you do when you install something on your own server, and it can look kind of unprofessional for your customers to have to go to another server to place an order, but it's definitely easier to set up.
If you want to learn to program your own shopping cart, by all means go for it! But if you don't have some background in it already, it's not going to happen fast enough to solve your problem right now. You're better off finding a ready-made solution for now, be it a script you install yourself, or a remotely hosted solution, and then working on building your skills up over the long run.