Most budget microphones are good for what you need them for, so you'll be fine with it. If your levels are too loud you have to monitor them before you hit record, keep some distance between you and the mic and adjust your mics level until you're peaking at about -10dB. Don't record your vocals loud, you want to have nice clean vocals that aren't clipping and you need headroom. If your vocals are too low in certain areas I'd recommend some careful automation, and once you get comfortable learn about compression. You can use the compressor to bring the loud areas to a relative level with the lower areas, but don't use it unless you know what you're doing, you'll end up doing more bad than good.
Audacity was the first piece of software I recorded vocals into, I don't recommend it. I'd recommend you use something like Reaper, you can evaluate the software and when you're ready to purchase a license for the software it's the cheapest option out there. Exporting your takes individually is going to be dependent on the software you use, you'll just have to search that up.