From Premier Inn T2/3 to T3 any bus that takes you to Heathrow Central Bus Station will work, so in addition to 105/111 mentioned by KARFA you can add 278, 285 and 140 (and for very early departures N9/N140 are night buses, although that shouldn't affect you). There should only be a few minutes between buses.

If you are booked in the Premier Inn T5 then you have two options to get to T3 - either walk about 8 minutes and catch the U3 direct (simpler to walk to Pinglestone Close), or no walk then 81/423 to Bolton's Lane, cross the road (there's a lights controlled pedestrian crossing) then any of the buses above to Heathrow Central. You will need to touch in again on the second bus but if you use the same card you won't be charged as it'll be under an hour since your first journey started.

Allow 10 minutes to walk from the bus station to T3 departures - there are moving walkways. At the final stage you have the choice of a short escalator or lifts - unless you have a lot of luggage, the escalator will be much quicker.