If a contract is signed for reserve service while still on an active duty contract does the reserve service start immediately following ETS?

Your contract starts the day after ETS. You can also cancel the contract at any time if you change your mind and choose to stay on active duty

Since your question was already answered, I just want to add something.

Try and get a POC from your career counselor before your ETS.

Reach out to the unit and ask for all necessary in-processing information you may need, and a current BA schedule.

Don't just blindly show up to a unit that's hundreds of miles away without getting in touch with someone or getting dates and schedules.

Yes but the drill orders should arrive before your ETS so you know the where and went you have to report. RIU (Report in Uniform) but most times you can 'informally' report in ahead of the drill to get your paperwork up to snuff. Good Luck.