Plantar Fasciitis - The Plantar Fascia is a band of tissue that stretches over the arch of your foot, from your heel to your toe. It is often mistaken for Heel Spurs because it feels like a bruise on your heel. After walking or running around for a bit, the pain often subsides, but the first several steps after sitting or laying down for a while are very painful.
The best treatment for them seems to be foot massages (or rolling foot over something round like a tennis ball or a water bottle). Icing the area may help. Also, this problem often occurs when shoes are well worn. Replacing shoes can often help. Inserts in your shoes that fit your arches well are often recommended, but that is a matter of opinion, and I don't necessarily recommend it.
Achilles Tendinitis - This is not something I have experienced, but it appears that ice is probably the best treatment. Stretching is also advised, but you may want to study up on proper stretching. Overstretching could make it worse.
Ankle Sprain - Should you roll you ankle on a rock, root, curb, etc., the answer to when you should start running on it again is dependent upon the severity of the sprain. Definitely you will want to practice the R.I.C.E. treatment. Then, if you are able to put your weight on it, try taking the advice given in "Stop or Run Through It?"