Due to lack of gaming systems, I'm mostly a web/PC gamer right now. But I'd still really like to try some of the classic Mario games (New Super Mario Bros, SMW, SM64, etc), and I know that emulators are allowed for most of these. So are there any emulators that get used frequently or seem to be better than the others? Thanks in advance.

You’ll have to ask the mods of the games. Most games have approved emulator versions and configurations and they differ game by game, there is no standard.

Often times the rules for each game will also specify the allowed emulator and any special rules for using them.

BizHawk for basically everything. Other than that, Project 64 1.6 is my favorite N64 emulator. Version 2.X is banned

Although it is just a lot better to ask the moderators, I'm just telling you what I use

Bizhawk is actually quite often banned in my experience, so not really the best one for speedrunning.

If you get to SML and SML2; BGB and Gambatte are our emu standards.

I can speak for NES your go to is Mesen, followed by PuNES, Nestopia UE, BizHawk, FCEUX