There are a few options open to you depending on what path you want to down (hardware/software).
If you are looking to really learn how to beat match then there are a couple of options.
1) Get yourself some second hand turntables (vinyl), a 2 channel mixer and some speakers.
Gemini are a cheap, beginners brand that should do the trick (I started off on geminis) but they do however lack the build quality of the more expensive turntables
2) CD turntables (http://www.decks.co.uk/products/cdplayers)
3) MP3 turntables (such as traktor by native instruments http://www.native-instruments.com/#/en/products/dj/traktor/) (note: you will need a laptop/pc for this also as well as a midi controller)
I have a slightly biased opinion on this as I started off on vinyl but I do think you get the 'feel' the music more with vinyl (but this is just an opinion). More and more people are turning to 3rd option with products such as traktor (which does look amazing) but the software can help you 'auto beat match' tracks which takes away an element of the process. Some people argue that it allows you to concentrate more on the mixing and creating an atmosphere, some people argue it takes away an element of skill.
With the 3 options above, you would need to know the basics of beat matching and dance music composition. The latter is relatively easy as you only need to know a snippet (i.e. there are usually 4 beats to a bar and after 8 bars a new element comes in on the track). The first part takes time and patience. The idea is to layer up to tracks together, one will be played via the speakers and the other will be played via the headphones (usually). The idea is to listen to the two tracks, one ear listening to the track through the speakers, the other ear listens to the track through the headphones and then you have to determine which track is the slowest/fastest.
The reason I like vinyl in this respect is when is when you are cuing the track in the headphones you can 'feel' the start of the beat through your fingertips. So you cue the track and then layer it on top of the other, and if the track in the headphones is faster (to learn this just takes practice, you can't really teach it) you need to slow it down using the pitch sliders on the sides of the turntables and if it is faster you need to speed it up.
When both tracks are in sync (i.e. have the same tempo) you can then start to mix them together via the speakers using the mixer. I rewind the vinyl and count the bars and when the new element on the track in the speakers comes in I layer the track in the heaphones on top of that. This ensures that when something new comes in on one track, it comes in at the same time on the other. Slowly then bring the crossfader into the middle and then increase the volume of the other channel so both tracks can be heard through the speakers at the same time. It is your choice how creative you want to be in this stage. The goal is to fade out the other track and blend into the new track seamlessly.
This does take time and practice so be patient and just enjoy it and experiment with it.
Hope this helps.