My understanding is that the new battletech manual is a mech-only rulebook that pulls the relevant parts from Total Warfare, Tech Manual, and bits of TacOps. It does not include combined arms rules.
TW has all the gameplay rules for vehicles, infantry, aerospace, and civilian equipment.
TechManual is the rule book for building all the unit types covered in TW.
TacOps includes advanced rules and equipment for the standard scale game.
StratOps is advanced naval combat and construction, some maintenance rules, and battleforce, which is a quick play system for large unit actions (handles 36v36 in the time the standard rules handles a 4v4). Alpha strike is a derivative of battle force, using the same quick play system but at small unit scale.
As to my opinion on what to play: Classic has a ton of bookkeeping required, which slows play down a lot. It is very detailed and has a lot of depth, but that bookkeeping overhead is a barrier to entry. I’ve found that I enjoy it more in MegaMek, where the program takes care of that overhead for you.
I’ve not done alpha strike, but I’ve done my research. It seems that it plays more like a modern tabletop but keeps hallmark elements of battletech. Resolution is much faster, with less rolls and less tables to consult/memorize.
Ultimately, it comes down to what your play group is down for.