Powering a 30 watt peltier heatpump
Use full wave rectification, not half wave. HW uses transformer poorly, may not be good or TEC, has no obvious advantages except the cost of 3 diodes.

If you want to operate it at full power with no control of cooling level then 12V is fine. LM350 regulator needs about 3V headroom. So 12V out from regulator = 15VDC in min.

Full wave rectified transformer will give ABOUT Vmax DC ~=  1.4 x AC voltage.
 Or VAC_min  ~= (Vdc + dropout) / 1.4 
 So 12V + 3V = 15V
 VACmin ~~~~= (12 + 3) / 1.4 =~ 11 VAC if ~= NO ripple voltage
 ie 12 VAC transformer will give 12 VDC after regulator if well smoothed.
 More is better, so maybe 14 VAC - 15VAC will allow regulator headroom plus some ripple allowance.

If all you want is to run it at full power then a transformer, bridge rectifier, smoothing capacitor and series resistor are all that is needed. Resistor drops excess voltage. A 10 VAC transformer will be about enough (1-VAC x 1.4 = 14 V with smoothing and ripple) and 12 VAC will definitely be enough. Resistor dissipates 2.5 Watts per volt of drop. A length of Nichrome wire adjusted to provide correct voltage to Peltier is one option or 0.4 Ohms per volt of drop select-on-test. 

Regulator WILL need heat sink - how much depends on transformer. At say 3V regulator drop - about the minimum you should figure on, the dissipation  V x I = 3V x 2.5A = 7.5W.  Say allow 10 Watts. More if transformer is of excessive Voltage.

Heatsink can be selected using degree C (or K) per Watt for commercial heatsinks. For a 10C rise at 10 Watts you need 1 C/W heatsink which is "very large indeed".
  If you want heatsink at almost cool enough to touch (almost) say 60 C the if ambient = 30 C worst case heasting delta T = 60-30 = 30C so heatsink = 30C/10W = 3 C/W
 Even that is largish. Going other way, 10C/W is common so 10 W = 10W x 10 C/w rise = 100C./ Add ambient + Tsink = 100C rise + 30C = 130 C.
 You rally don't want 130 C heatsinks.
 so somewhere between 3 C/w and 10 C/W leaning towards 3 C/W end.

Fans and Peltier together are OK. Fan load is small compared to Peltier load.
Fans could run from smoothed DC before regulator - maybe with a dropping resistor of their own suited to VFan an Vdc.

Strip board construction OK but keep wires short and heavy. If running higher currents along a piece of stripboard you can solder wire to the strip for longer high current leads or use a wire link from points to be joined.  

Main mains caution is DON'T PLAY WITH MAINS whn not needed. eg here all the circuitry is LV apart from mains feed to transformer primary. Do the primary side wiring well. Insulate as required. Then leave it alone.