Mics are the most important part, but a good mic will be absolutely useless unless you have an interface. Plugging your headphones or microphone directly into the microphone and headphone jacks on your computer will create a lot of noise during recording and playback, mostly due to the low quality jacks and preamps. They work fine for just casual stuff like Skype chats or listening to music, but for any type of voice acting or recording for music or anything similar they are no good, and it won't matter how good your microphone or headphones or speakers are.
Interfaces are usually USB, (though there are others like firewire with less latency) and they'll give you much better sounding recording and playback. Cheaper ones like the Presonus Audiobox and the Scarlett Solo are only around $100 USD but you still need a microphone to go with them. Seeing as you are on a budget, you should look into USB microphones with built in interfaces.
The cheapest I've found with good reviews for recording voice is the Samson Meteor, which is around $70 USD.
The Blue Yeti also has a built-in interface for $120 USD I believe, but I've had a bad track record with Blue microphones myself.
Headphones are secondary mostly when recording, but still important. If you can, buy a cheap pair of reference headphones (you can get them for around $20 - $30 USD). Don't worry about having great headphones right off the bat, what you have now will probably work fine. Having a decent microphone is the most important.
Pop filters are also necessary. You can find these on eBay for as little as $3 (that's where I got mine) but at the end of the day its just a piece of mesh fabric. I've seen people make them with everything from coffee filters to panty hose strung up with wire hangers. I got by originally by draping a scarf over my microphone.
As far as software goes, Audacity is honestly amazing, especially for voice acting. It should work perfect, and it's free. It has VST support, too, in case you want to use any effects or something similar.
tl;dr- Plugging a microphone directly into your computer will sound terrible regardless of how good your microphone is, so buy a USB interface, or as a cheaper option, a USB microphone with a built-in interface.
Some good things to look at:
https://www.youtube.com/watch?v=y4pW0Cvzw3g - Microphone guide
https://www.youtube.com/watch?v=UO30QsZG1Cg - Interface guide
http://mylittleremix.com/viewtopic.php?f=13&t=695 - Another guide to microphones, covers USB microphones and interfaces
http://mylittleremix.com/viewtopic.php?f=33&t=855 - Singing guide, but some techniques may translate well to voice acting