I run a combat-heavy game. I used to run other combat-heavy games (or at least, combat-critical games, where combats could make or break you). Fighting is central to my GURPS experience, and GMing combats is central to my GMing experience.
So here are some ways players can help speed along combat. The more of these you can do, the smoother fights go.
Know When You Go. However the GM is doing turn order (by Speed, round the table, by bribe size, etc.) know when you go. This alone will speed up combat significantly over "sorry man, I didn't realize I go after he does."
Know Your Best Options. You don't need to know what your modifiers for doing everything are, just your modifiers for doing the stuff you do a lot. Put them together in a way that's easy to parse out but also easy to total at a glance. If you just write "Broadsword Swing to Neck 15" and have to re-calculate each turn to know that you have to add +1 for your Balanced sword, it's not really faster. One way I do this is to spell it out with a final number.
For example, I might write: Tarjan's Twofer (Rapid Strike (-3/-3) to Skull (-3 w/Slayer Training), net -6, +1 for Trademark Move = -5/-5) and just know it's effective skill -5/-5 for this move (thanks to two different advantages and one perk), or I might write that plus "= 17" at the end thanks to my One-Handed Flail-21 and a +1 for a Balanced morningstar.
Know Your Situational Bonuses. There aren't a lot of these, but they exist. Know that Retreat gives a +3 to Dodge and a +1 to most other defenses (a couple skills get a +3 - if you have one, know this). Know you can't do it repeatedly. Know your DB (defensive bonus) and when it adds to your defenses (front arc plus one side for shields, all arcs for most magical DB, for example).
Know Your Defense Cascades. By Defense Cascades, I mean, your penalties for cumulative defenses. Dodges are unlimited, at -0. Parries are at a cumulative -4, -2 if you've got Weapon Master or a Fencing Weapon or certain two-handed weapons (using the optional but highly recommend rules in GURPS Martail Arts), -1 if you meet both criteria. Block is -5 for each additional, again halved with the appropriate Weapon Master. Write these down or at least write down the penalty. If you're always rolling against a net Parry 17, +1 with Retreat, you could write "Sword Parry 17, +1 with Retreat, -2 after first." But know it, don't calculate it each time.
Know Your Damage. You shouldn't need to look it up, figure it out and write it down. If you do a move that adds extra, well, add that in on top at the end.
Know the Page Refs For Your Crazy Moves. If you do unusual stuff, know how it works and have the page ref down if a critical edge case comes up - or just so you can flip to it before the game and hand it to the ref and say "Don't forget I'm probably doing disarms today." If you have specific penalties for special moves (double shooting, hurling handfuls of shuriken, doing twofers with your staff) note them down along with the page refs. I prefer people don't look stuff up in play, but we're going to look it up eventually, so know where it is.
Remember What You Did Last Turn. Don't forget that you attacked last turn with your unbalanced weapon and need to re-ready now. Especially don't remember this after you parried with it anyway. Don't forget that what you did last time affects what you can do now and right up to now. If you know you'll forget this a lot, do everyone a favor and get weapons that can both attack and parry on the same turn and which don't require time to ready!
Those are some easy ways to keep combat moving along that the players can do.
Short version - know your capabilities and your rolls, and be ready when it's your turn!
A later addition: Part II