Press the button on the left-hand side of the screen:
The main tutorial introduces you to the 8051, and there is an 8052 tutorial that explains the differences.

To be precise, neither of them!
The 8051 & 8052 were chips introduced by Intel some thirty years ago - so you should certainly not be using one of them if you're just starting out today!
However, there is now a whole vast family of literally hundreds (possibly even thousands?) of derivatives that are built upon the 8051/52 architecture. All of these derivatives share the same processor "core", and just add different sets of "peripherals" around it - so once you have learned the 8051 "core", you can program any derivative!

That is true.
EPROM is 30-year-old technology contemporary with the original 8051/51 - so, again, you shold definitely not choose to start now with EPROMS!

You should look for a modern 8051/52 derivatives with on-chip Flash code storage that can be programmed with the chip in your board ("In-System Programmable" or ISP).

Try to get yourself a simple, ready-made, starter kit - like this one:
http://www.8052.com/forum/read.phtml?id=77698

Also look at the Silicon Labs (formerly Cygnal) kits - these have the great advantage of on-chip debugging hardware!