If I do not get my vehicle serviced at a dealership, how will I be able to prove I have done the recommended maintenance on my car?

Keep dated records and receipts for every service, with an accurate description of the parts used and work performed. And remember the burden of proof runs the other way: if a manufacturer or dealer wants to deny a claim because you used an independent service or a non-OEM part, they have to prove that part or service caused the failure. It's not on you to prove it didn't.