I've used spivak and munkres, lee's trilogy, wald's GR, and now kobiyahsi nomizu. I'm working on general relativity from a math standpoint. I think the big shift is taking things from the standpoint of structures and actions rather than as vector fields qua vector fields is much nicer and makes the math easier.
The wedge product is just an algebraic nicety though it does have all the properties of the cross product, like volume etc.
Coordinates are hard to use from a calculus standpoint, but kobiyahsi nomizu makes them really simple to define and work with. I would recommend forgetting about coordinates and just working on differential geometry from a connection standpoint for a while.