Heathrow Airport-transferrring terminals
 

How does one get from Terminal 3 to Terminal 2 and Terminal 3 to Terminal 4 at Heathrow Airport?

Easiest way is HEX (Heathrow Express). It's free between Terminals.

If you go to the Heathrow website they have some pretty helpful PDFs explaining transfers between the various terminals. I printed it out and found it helpful for planning purposes, though everything is such a mess with construction at that airport that the PDFs weren't really accurate anymore.

The best thing to do is plan on something like 1.5 hrs to transfer, because--esp. with construction--you have to walk very long distances. Also expect to ask people working there--due to construction the signage was less than ideal, and I ended up needing to ask for directions.

I assumed too much when I answered with HEX. Do go to www.heathrowairport.com for detailed transfer information.

HEX doesn't got to T4 - well not after the next couple of days

How it's done depends on if OP is doing it airside or landside

Airside it's a case of following the signs to "Flight Connections Centre" when they get off the plane.

Landside -

T3 to T2 : follow the signs, it's a bit of a walk.

T3 - T4 : follow the signs to Heathrow Express / Heathrow Connect and take the Heathrow CONNECT to T4. It's free between terminals