Do I need to buy a socket board? Which one?
You don't need to buy a socket board, but that's probably the most convenient way to work with TINI initially. The TINI board is a standard 72-pin SIMM. You need to feed it power, send it data, read data from it, etc. The only way to do this is to wire it up. You can solder directly to the pads (not recommended!), or buy a SIMM socket (e.g. AMP part #7-382486-2 or equivalent) which can be stuck into a breadboard (you will also need a regulated power supply, connectors, etc.). Alternatively, you can buy the socket board which provides you with the socket and many other connectors, plus external circuitry.
The socket board comes either with or without a voltage regulator. If you choose the board without the regulator (the TINI Sockets E10), you will need to feed your TINI a maximum of 250mA @ 5.00 V +/- 5% - in other words, you need a regulated power supply (an unregulated transformer will definitely not do.) If you choose the board with the regulator (the TINI Sockets E20 or E50), you can plug in any power supply, AC or DC, between 8 V and 24 V. Chances are, you already have one with the right voltage and the right connector from an old external modem or telephone answering machine. It needs to be rated at least 250mA for the SIMM, more for the socket board.