Some people will tell you that mixing is an art. They are wrong. Mixing is 100% science and technology used to blend electrical signals to create a mix of audible signals.
Good mixing is an art.
This is the ability to use that technology to create a mix that's pleasant to the ear.
My recommendations will likely be thrown out by others, but it will give you a chance to learn some basic at little to no cost while you work on the purchase of your equipment.
To learn mixing I recommend you first start recording yourself and seeing what constitues a good mix. To do this just record karaoke tracks with yourself singing and play them back to see the results. Apply effects and see how they change the music. To do this you need the computer you already have (you're posting here aren't you?), Audigy (a free software program available on the Internet), a microphone and a pair of speakers compatible with your computer. This will allow you to get an understanding of the basics of mixing. As you add equipment use that to feed the recording so you're using the same equipment you'll use at the show. Once you have all your equipment try inviting some friends over for a night of karaoke (a small group) to get a feel for finding discs, etc.
Assiting someone or even watching them (ask permission to sit close and watch if you can) can help. If they make a change on the board listen for the cooresponding change and think about if it sounds better or worse. If you can spend time with someone behind a board asking questions then jump at the chance.
That aside, I recommend the Basic series by Paul White. They are small pocket books that discuss basic techniques for effects, mixing, live sound, and other music related technical information. They are inexpensive and can give you an idea of some of the why's so you don't ask as many questions when you get a chance to work with someone behind a board.
Remember that disc management is key to being able to finding the right disc, right away. I pull a disc, que it up, and when a disc is done I carefully put it away. I never let a disc sit out where it can get knocked off the table or misplaced. All my discs are in order by manufacturer/disc number (ie. SC7501) the same way they are labeled in my books. This makes looking up a disc VERY easy for me. I'll likely be moving to a different storage mechanism for my discs soon and PC based hosting in the next 6 months or so. I found that using slim line CD cases w/ a label w/ the disc number on the front made picking discs easy but takes up some room. Books are harder to handle for a show but possible and I see plenty of shows that use them (I prefer better protection for my investment than books).
As for equipment:
Most often you'll find a variety of recommendations and here is no different. Basic recommendations will be some of the items below that have a proven track record and are able to withstand the rigors of gigging week after week. I recommend you listen to the recommendations, read the reviews, find a place to see the equipment first hand and use it if possible and then purchse. Make your own informed decisions instead of listening to what we recommend and blindly buying it.
1. CDG Player
I recommend the RSQ/JVC 333 - 3 tray player. It's inexpensive, I can que additional discs in each player and mine has been reliable. The single tray version - the 222 is also recommended quite often as reliable. Other options are the CAVS 99/199/202G that some people prefer or PC based hosting.
2. Mixer
The most often recommended is the Mackie DFX-12 for it's cost and built in effects. I use a Soundcraft Compact 10 but it's a recording mixer not well suited for live sound. Behringer is often recommended but extensive discussions regarding the reliability are often mixed (pun intended).
3. Microphones
I use Fender P-51's. They're cheap (under $50 each) and hold up well to being dropped, etc. They do require more gain than many others though. Shure SM58's are highly recommended and seem to hold up well to the rigors of karaoke shows ($100 each). The Sennheiser e835 has been discussed quite regularly and most seem to like the mic a lot ($100 each or $200 for 3).
I use a wireless mic for annoucing occasionally and there are a number of recommendations along these lines. I'll let someone else make recommendations here since I'm still doing research and haven't bought anything I'd even remotely consider recommending (hint: don't buy $30 wireless mics off eBay).
4. Amp/Speaker
Two different ways to go - powered speakers or amp+speakers. I'm going with powered speakers - either the Mackie SRM450 or the JBL Eon G2 15. Both have recieved excellent reviews on the board and from the people I know that use them regularly. These are both powered speakers.
Some people use sepereate amps/speakers and I'll have to let them discuss the specifics since I haven't done much research in this area because I'm not planning on purchasing similar equipment in the near future.
5. Cables
Use good cables and have spares. If you have problems with ground loops then feel free to ask here about resolving them - I recommend Radio Shacks Ground Loop isolator for unbalanced connections and it's inexpensive at $15 each. I keep one around to help weed out problems. You'll need more extension cords and power strips than you think you will.
6. Lighting
I have a small desk lamp and flashlight I bring with me to help find things in the dark after the show starts. It clips onto the table or my rack and is non intrusive.
7. TV
8. Stand(s)
These are all personal choices and pretty intuitive.