I'm just starting to learn C++, and I was wondering if anyone could suggest a few small projects that I could work on that'll help me learn more about what I'm doing.
So far the most complex thing I've written is a program that runs when I first log in to my computer (yay auto-run) and writes the date/time to a .txt file. So I'm looking to move a step up from that. :-p.
Take a look at codekata, I haven't tried it myself but I've heard great things, and it looks like a great way to improve your programming skills.
What will hold your interest so you're motivated to keep working, even when it sucks, is going to depend on you. What will help you get to where you want to go will, again, depend on you. Sorry, we just can't answer this one for you. The best advice I can give you here is practice, do things you find interesting and/or useful, practice some more, and did I mention practice?
That said, if you want to test your knowledge of syntax (which is something you just have to learn in any language), try the exercises in your C++ books. (Or visit a book store for a few hours and pick out some that look like they will help you.) Those exercises are designed to teach the very basics such as syntax, and answers are often provided. Hang around on SO and answer some questions too. :)
Games, write simple games.
Start with very simple ones, like tic-tac-toe, hang man, graduating up to more complex interactive games like snake, or battleship.
Even with something simple like a snake game, once you've the initial game together, you can add extra computer controlled snakes, add boundaries, add moving boundaries etc etc.
Games are an excellent option, you don't need to learn specific technologies, you don't need advanced physics engines, you don't need to learn the ins and outs of an obscure business domain.
Writing simple games are an excellent method to exercise problem solving and practise writing code.
Just to state again, incase there's any confusion, I'd recomment writing games :)