I've been finding the best way to save the game is to quit playing after a cinematic clip. Once you've watched the clip the save icon starts rotating in the right-hand corner. You'll have to rewatch the clip later, but at least the game saves up to that point (instead of having to replay the level form the starting position).
Not having taken my save file online yet, I couldn't tell you whether the offline data syncs up with the online community. Although, I think I've read somewhere that it does... as to tie up with some of the trophies, such as upgrading weapons and gathering one-million bolts. I'm thinking the save data should match up, but for the time being I've been working away the offline experience first. Next up: multiplayer.
Checkpoints are not counted as "saves." Saving the game happens whenever you're looking at the spinning gear icon at the bottom-right of the screen. At least, that's the way it goes sometimes. I've seen the icon spinning and had the game reload again at the start of the stage or a couple of segments back. This auto-saving mechanic is not ideal, though it is better than not having one.
Having experienced some of the online aspect now, I can confirm that your save data does transfer directly to the multiplayer and then back again to offline. If say you play through the offline together with your friends, all your separate profiles, bolt counts, and gadgets/weaponry should once again be secured for network usage.
