if the starter was turning over its not a starter issue, fans would not be a problem at all, they come on when the car is on and at a certain temp (if u really want to check them run the car for a while and then b4 turning it off check and see if they are running, if ur car isnt over heating should be something to be worried about), id suggest a full tuneup, ie spark plugs, plug wires, cap and rotor, make sure the battery is fully charged (if not any local parts store will charge it for free) change ur oil, basic car maintancance is probably all u need, and YES check ur transmission fluid, probably get it replaced with new atf, vibrations occur because either the engine isnt running the consistently on all cylinders, on any new car id suggest replacing all ur basic maintance replacement parts, check ur belts for wear, possibly also replace fuel system components at very least the fuel filter and pcv valve, air filter etc. good luck with the car and let me know if u need any more help