Hacking in normal horsey terms is going out for a ride (off the premises of the riding school - i.e out in t'countryside)
Livery is the horse's keep - their stabling, feeding etc. If a horse is in full livery, it means someone else is doing the hard stuff for you. DIY livery is doing it all yourself, but at someone else's premises.
Common ridings - have no idea (never come across that before). Is the yard near a common? Maybe your Riding school uses that expression to mean riding on the common as in away from the riding school, but not in open countryside.
Tie ups - again, don't know if this is a riding school expression (my field of expertise is racing). In racing, a horse 'tying up' means they have suffered a muscle spasm which makes it difficult to walk due to stiffness, and can't be ridden (it is a temporary thing, like cramp)
Tack is all your saddle, bridle and bits and pieces.
The gaits are walk - trot-canter-gallop. Ideally you should rise to the trot immediately, but sit down deep into the saddle (sitting trot) just before you canter.
Don't worry, most riding schools know this is supposed to be fun and if it seems a bit too serious and hostile (sadly, some are) then find a different riding school that doesn't take itself too seriously. DD1's was lovely and friendly and very relaxed (which is how they all should be)