If you consider yourself handy, then you're probably capable of solving minor plumbing problems yourself. You can plunge the toilet, clean the drain trap, and maybe even install a new faucet. It's nice not having to call the plumber for every little thing, but you should certainly always have a plumber's number on-hand. These problems are ones you should leave to the pros.
1. Leaks of any size.
If you come across a tiny leak coming from a little slit in the pipe or a loose connection, you might figure you can fix it yourself because it's small. But even the tiniest leak usually requires that a section of pipe be replaced. Makeshift solutions, such as putting some tape or plumber's putty over the leak, will only hold up for a couple of weeks, if that. Replacing pipe is a big project that involves cutting metal and making new connections, and for that reason, it's a project to be handled by a plumber.
2. Multiple slow drains.
If one drain is slow, you can plunge it or perhaps pour some boiling water down it, and that should address the issue well enough. But if several of your drains are slow, the clog is likely located into a larger pipe that connects all of those drains. You can't reach this clog without major equipment, such as a plumber's snake or a hydrojet, and neither of those pieces of equipment is safe to use without training. Not to mention, buying either one will cost you more than hiring a plumber.
3. Corroded water main valve.
The main water valve is arguably the most common valve in your plumbing system. If you turn it off, you'll shut off the flow of water into the whole home — which can be lifesaving if you have a leak. If this valve is so corroded or covered in a mineral buildup that you cannot turn it, then you need to have a plumber come out and replace it. They will need to remove the existing valve and put a new one in place without letting water gush all over, which requires a lot of precision and special equipment.
Basic DIY skills can certainly come in handy as a homeowner, and it's really respectable to solve your own minor problems. But don't try to fix a leak, a major blockage, or a corroded water main valve yourself. Call a plumber.