No particular order, I normally start with whatever is causing a problem or that I have noticed requires attention. Having recently done some servicing on my bike including replacing gear cables and brake pads I've noted some of the things that over time I've paid insufficient attention to:
Clean underneath bottom bracket
Clean & lube F & R brake pivot points
Lube & turn F & R brake barrel adjusters
Clean & lube F & R derailleur pivot points
Lube & turn R derailleur barrel adjuster
Lube & turn downtube barrel adjusters
As I tend not to use the rear brake so much, and rarely the front derailleur, these were clean on the outside but had got clogged quite badly internally over time. The front derailleur and associated barrel adjuster wouldn't shift / turn at all. Now stripped, cleaned, reassembled, reindexed and working perfectly, but I'll put these on a checklist so that they don't get into the same state again.