Is a Nikon D60 with 18-55mm lens a good, reliable camera for a beginner?

That's a fine camera. It was Nikon's entry-level model when it came out in 2008, and it replaced the D40 and D40X, the former of which was a top-selling DSLR for many years straight.

In general, it's plenty reliable, but a) like all DSLRs, it does have a lot of moving parts and b) like all mass-produced electronics, there are sometimes duds. Are you experiencing a particular problem that makes you ask this? If not, don't worry.

It's also several years out of date. Nothing wrong with that, but in particular, its resale value will be low. It's still a capable camera, though, so unless money is no object, it doesn't make much sense to sell it. You should use it to learn the basics, and when you get to the point where you are ready for something else, you will know — and then you could pass it along to someone else and continue the favor you've received.

The D60 was replaced by two lines, the more-basic D3xxx line and the slightly higher D5xxx line. One can get the D3200 for under $500 with lens, and the D3300 for under $600. And I see used D3000s going for as low as $100, so I don't think you'd get more than that selling yours unless you sucker someone. But all of these cameras are perfectly capable despite their age, and in some sense worth much more than they can be sold for. So, again, I'd really suggest getting the value you can out of it, unless you already know that you're ready to put serious money into photography. In which case you might want to start with a newer body, which will be faster, have a nicer LCD screen, and be better in low light. And also in that case, upgraded lenses — a whole world to learn in itself.