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

You need to see if there is a Reserve or National Guard PAC/S1 near, Many of them can input DEERS.

You can do it this way but it will not be easy as it could if you planned ahead. For example, if you have a wedding planned during leave that's fine but if you went ahead and jogged over to the court house one day and got married you could get your paperwork done in advance. After that you can get your marriage entered into the system and add your spouse into DEERs. You arriving to your duty station with your profile up to date will allow them to better assist you. Just saying.

There is no manic bullet that will speed the process. Too many factors involved. Prior planning was the way to go. You need to submit all of your documentation at your unit. You also need to tell your spouse to be patient. Depending on where you are going will largely impact the requirements. Oversees requires health examinations and to determine if your spouse is An EFMP or not, which in turn determines whether or not the spouse can join you on the assignment. Anyway, PPPPPPP.

Get your paperwork together.

It's not hard. Time consuming yes.
Marriage license, her DL, proof of residency, and where is she now. All things you're going to get asked for.

You should know this is always a bad choice and causes unnecessary hardship on the spouse and financial strain on you