We have discussed and re-cussed this topic many times. The problem is that the choices are almost entirely based on FUD and personal opinion. So, there are almost as many conflicting opinions as there are members of this forum; and very little consensus.
That said, here are my opinions, based on many personal experiences and hundreds of thousands of miles covered in a TR3/A under pretty harsh conditions:
Crankcase : Valvoline VR1 (racing) oil 20W50 unless you live in a colder climate and/or have a freshly rebuilt engine.
Gearbox & OD (if fitted) Redline MT-90
Rear axle and steering box : Valvoline SynPower 75W90 gear oil
Brakes and clutch: any fluid rated DOT 5 (not 5.1). Brand not important, I usually buy from TRF just because their price is good (and I like to support TRF).