What should I expect going to my next duty station after getting married while on PCS leave? What do I need to know about DEERS enrollment?

Bring your paperwork to the nearest military personnel office ASAP and get your new spouse registered and an ID card.

Where are you on leave and what's your next duty station? Enrolling in DEERS ASAP is critical. You don't need to be near your next duty station or a major installation, there are DEERS/TRICARE offices at tiny military activities all over the place.
1) Google "DEERS Office near me"
2) Call first to check their office hours -- They may not be open 9-5, M-F
3) Go to any ID card-issuing office to add your spouse to DEERS.
4) You'll need the following documents (originals or certified copies):
-- Marriage certificate
-- Spouse's birth certificate
-- Spouse's Social Security card
-- Spouse's photo ID
Do you have a sponsor at your gaining unit/installation? I'd contact the housing office at your next installation. They may have an advanced application or waiting list that you can get on before you report in.
Congratulations and Good Luck
Know Before You Go: https://www.tricare.mil/LifeEvents/Marriage

How far of a drive is it?