Not a 737 but still Boeing.
1. Cockpit setup
2. FMC route - normally it's ACARSed in so request it, go get coffee and it should be waiting when you get back. Otherwise type from the paperwork. If it did come from ACARS check each leg against the paper flight plan.
3. Get the ATIS
4. Pick a runway and star and load them. Compare what's in the FMC to the chart.
5. Get the takeoff performance. Most of the time it again comes through ACARS, otherwise we work through tables.
6. Get the ATC clearance. Make sure it is the same as what you loaded in the FMC.
7. Briefing. One time a Check Airman was giving OE to a new Captain and I was on the jumpseat. The new Captain was going on and on. The Check Airman was looking out the window and picking at his nails after three minutes. The new Captain just keep going on. When he finished and asked if there were any questions or anything to add the Check Airman pointed out it was supposed to be a briefing and not the complete FCOM. I cover:
a. Current weather
b. Any MEL items that matter
c. Taxi plan
d. Takeoff thrust settings
e. RTO considerations
f. If there is a special engine out procedure and if we'll need fuel dumping
g. The SID step by step
h. MSA and transition level
8. Preflight Checklist
If everything goes right it's about 15 minutes of work.