Toomek
M3 southbound is fine, the traffic is all going the other way. The M27 Eastbound up until 07:30 is usually no problem. It starts to build from then and some mornings can be stop start from Junction 5 to Junction 9 by 08:00. Low sun in the Spring slows it down.
Eagle112
It depends when you are travelling and where you are staying. There is a direct National Express coach from Heathrow to Fareham and door to door that is the quickest and cheapest. Unfortunately the departures are infrequent so you may have to wait and at peak times they can be badly affected by traffic.
The problem with the train is that Heathrow has no direct rail connection to the trains to the South. Off peak the Rail Air coach to Woking then train works well but the coach goes on the M25 so can get held up.