Oil change + replace battery, which comes first ?

Oil is probably fine. Change the battery and go camping. Have the oil tested if you have any doubts. Best to fully warm up the vehicle before you park it for an extended period. That means driving it 20+ minutes. No need to start it unless you will again be driving it to fully warm up. Idling in place does not count. Next figure out how to keep the battery charged while in storage. And yes if you change the oil then change the filter too. Check the tire pressure too before you roll.