Dont use the push in connections ever would be my recommendation and dont buy the outlets that have them in them. IMO they are the source of lots of problems down the road.
A pig tail is just a short wire used to connect several wired together with a Wire nut and then the other end of the pig tail has a loop bend to go under a screw head on an outlet or switch. It is a better method of connection and in the future if the outlet wears out all you have is two screws to deal with in replacement of the outlet. Also if an outlet stops working all the rest downstream will work fine.
If you dont have a bare copper ground wire you should have a GFCI outlet as the first one of the string of outlets. In the case of a GFCI you will have 4 wires connected to it. the screws at the top are labeled Line thats where the power comes in. the screws on the bottom covered with a strip of yellow tape when you open the GFCI package up are labeled load and those go off to the next outlet. With most GFCI outlets you dont bend a loop on the wire to go under the screw head, all you have to do is strip the wire straight and insert it in from the back similar to the push in connectors you mentioned only on GFCI you then tighten the screw and it locks tight on the wire. GFCIs allow two blacks and two whites to be attached to the load side. So in that case you could go off in two directions from the GFCI and they would all be protected.
If you use one that doesnt have the bare ground wire the GFCI will protect you and all the outlets on the string. In the package there are stickers that you are supposed to stick on the outlet cover to let people in the future know there is not the safety ground wire in the box.
Here is a link showing the pigtails.
http://www.handymanhowto.com/how-to-replace-a-worn-out-electrical-outlet-part-3/
Here is a link to the GFCI
http://communities.leviton.com/thread/1080