What exactly is a backstab connection

I have recently bought a house that was built in 1958, and have discovered that the wiring isn't all that great (in the outlet and light/fan fixture boxes, the insulation is cracked and brittle, and the sheathing on the wire is messy and sometimes simply falls apart (I feels like the old cloth type sheeth).), and also has no ground. We've had a new panel put in, and it's been grounded to the cold water pipe (is this still allowed under the code?). I'm an amature electrician and can handle most of the re-wiring, but I don't have time to go through the house and completely re-wire it from start to finish in one shot. I will have to do it a little at a time, as I have time to do it. I've noticed that they seemed to use a wide variety of gauged wire (from 14/2 to 10/2), and I plan to rewire using 12/2 on most light and outlet lines. I have heard that you should not run 12/2 wire from a 20 amp circuit in the panel to a junction or gang box, and then have 14/2 (i.e. old wire that hasn't been replaced, yet) branching off from this to other outlets/lights. Is this true? If so, it would seem that I need to work my way from the fixture/outlets back to the junction boxes, and then back to panel, rather than from the panel out to each junction box/fixture/outlet. (Did that make sense?) Also, if I'm working my way backwords to the panel, and I replace the wiring from a junction box to a string of outlets (but haven't had time to replace the wire from the junction box to the panel), what do I do with the ground wire in the new romex? All of the original gang and junction boxes are metal, so do I simply connect to the box, temporarily, until I can run the new wire back to the panel so I can properly ground it? THanks for any replies!

I need to upgrade my system from 60 amps to 100 or 200 amps. I just do not want to be taken to the cleaners for this upgrade. All I want to do right now is upgrade the service panel for 60 AMP fuses to 100 AMP or 200 AMP circuit breakers. Could someone provide an average price for this upgrade? Thanks for the assistance.