Got married recently, and was best man for my mate this weekend gone.
Organising stag do is a big one - get a list of people he wants to invite early and introduce yourself. Check with groom who he considers essential to come, then don't pander to the rest around dates, venues, costs etc. Don't feel it has to be the cliche of getting hammered in fancy dress then going to a strip club if that's really not the groom's thing. It feels like herding cats, expect a few people to grumble about where/when etc and expect some to drop out at the last minute, usually after you've paid for something and ask them for money. Ideally have people pay for as much of the big stuff directly (ie. not via you) if you can.
Traditionally the best man arranges the men's clothes - that may not be necessary but it can be good to go to the suit hire place with the groom and ensure he picks something suitable and that matches colours of bridesmaid's outfits, etc. Also offer to deal with returning suits, etc afterwards.
Get quite involved with the wedding and plan - if you don't know the parents, can be good to find some excuse to just drop by and say "hi" sometime before the wedding. Any special needs, disabled access/arrangements, etc you should know and make sure ushers in turn are informed. Ideally be well enough informed of what's going on and when that you can be point of contact with venue staff, photographer, etc so they they don't need to bother the couple on the day. As said, make sure things don't go wrong, and if they do keep it out of sight/mind of the important people.
In the run up to the wedding, be available to help - ideally keep the day before free as there's usually ferrying bits to the venue, running errands, etc. Everyone underestimates how much time they'll have so another pair of hands are always appreciated.
On the day - Make sure men's party is well turned out and deal with buttonholes, etc. Obviously make sure the groom gets there well ahead of time and is kept calm if necessary. Brief ushers, make sure they know what's going on, where people can/can't park, etc. There's often a rehearsal which you should be at just to know what you need to do, where to stand, etc. For churchy weddings, you'll deal with little things like making sure the couple have orders of service for hymns, etc.
For photos, you and the ushers help get the right people in the right place at the right time so that photos get done as efficiently as possible - then everyone can get on with enjoying themselves.
You may need to MC - to announce the couple coming in for dinner, introducing speeches, cutting of the cake, etc. Again, discuss with wedding party what's needed.
Speech - start writing things down now. Anything that comes into your head that you might possibly use, write down. A page of a notepad, a note on your phone or a text file on your computer. If you leave it till the week before you'll draw a blank. DON'T swipe lines or jokes from the internet (or if you must, keep it to a minimum), you might not have heard them but most people have. Keep it all suitable for the audience (there's likely to be kids and/or old folk). Your speech comes last so keep it brief, people have sat through two or more speeches already - edit lots, and check you're not overlapping with FoB or groom. The only things you really need are to thank the groom on behalf of the bridesmaids (he should be toasting them at the end of his speech) and toast the couple at the end. In between, a bit about how you know the groom, what you've been through together, a little about how they met (although the groom will likely cover this himself) or how he told you about her and how happy your friend is with her. Be funny if you want, but no in-jokes (I'm sure it was HILARIOUS for the three of you who were actually there) or long and complicated stories.
Finally, you normally pay the band or DJ, often after they've finished. So don't get too hammered after the speech! You may also be required to dance with the chief bridesmaid after the couple's first dance (esp if they're not coming with a partner).
The stag do and speech are the "visible" things, but the preparation and keeping everything else going is at least as much work IMO.