You can root your phone without installing a new ROM. The kernel is the just core part of the operating system and that's the only bit you need to replace to root your phone.
A custom ROM is the whole operating system with the kernel and all the associated system binaries and applications. I'd stick with rooting first, which is lower risk. You can start to think about installing new ROMs when you are a bit more familiar with what you are doing.
Android users don't need something like Cydia or rooting to install apps from other sources though. We can just tick an option in the settings and we can install any app we can copy onto or download to the phone. It's much easier than with an iPhone in that respect.