I'll assume you've never done any concrete work before...
That said, the BEST thing you could do is find out if you have a friend, neighbor and/or relative who you could pay to help you. This isn't a difficult job, but must be done correctly or the end result will be worse than when you started. So...
To remove the sunken slab, you'll need to pry up a corner (with a long heavy bar & wood blocks) and break it apart with a sledge hammer. Get if off the ground or you won't be able to break it.
Build forms, using 2x4s with "form stakes." Match one end of the forms to the pre-existing sidewalk. Make sure they're level from the stoop to the old sidewalk, but have one side slightly
lower than the other (a perfectly flat & level sidewalk doesn't drain water). If it were my project, I'd put an "expansion joint" on the stoop end of the new sidewalk slab.
Pour some heavy-grade gravel into where you'll pour your pad. Level it out and tamp it down good. Without this base, your new pad will sink just like the old one.
For mixing the concrete, I'd suggest renting a small portable mixer like this. It will cost a little ($50 maybe), but will be worth it. http://www.quikrete.com/productlines/ConcreteMix.asp
Each bag will have instructions regarding how to calculate the amount needed. Pick up and extra bag or two. You can always take them back, but you do not want to run out in the middle of the job.
Mix 1-2 bags at a time. Use your garden hose and spray in water while the mixer is running. It is hard to describe the right consistency of the concrete, without calling it a "slump test," but roughly speaking, it should be about the consistency of pudding (maybe a bit stiffer). Smooth each "dump" of concrete with a small shovel until you have all the concrete you need in the forms.
Run a "screed board" (probably a 4' 2x4) across the forms to do the first leveling of the concrete. Fill in any low spots & re-screed.
If you don't have any kind of float, and don't want to buy one, you can use a short 2x4 (maybe about 1' in length) for the next smoothing.
After the concrete has set for a few minutes, use a hand trowel to finish the smoothing process. There is no magic to this, other than keeping the front edge of the trowel a bit higher than the back edge (not doing this will make you dig into the concrete).
For the best finished look, you'll want to "edge" your concrete as well.
It's kind of hard to describe, but not terribly difficult to do - assuming you take your time and carefully do things right.