Watson

Normals
6A - Overhead. Safe on block. Can be linked into 5A against crouching opponents.
2B - Two hit low attack, gives decent frame advantage on block.
2C - Launcher, and fairly good Anti-Air. Can be jump canceled on hit. Can be tech rolled. Very, very useful normal.
5D - Blanka-Ball style attack, strongest damage normal. Extremely unsafe on block, but can be made safe with traps or bubbles.

j.2B - Butt bounce. On hit or block, you will bounce and be able to do something else. Very useful for combos and other strange things.
j.D - Broom Ride style move. Hits overhead and can be TKed. On hit or block, you can cancel it into other moves (like j.B or 214K)
Chains
5A 5A ...
5A 5B 2C
5A 2B

2A 2A ...
2A 2B
2A 2D

5C 5D

j.C j.2B
j.D j.2B
Level Up System
Watson has the ability to level up by hitting 3 times with any of these normals: 5C, 2C, j.C
Leveling up will improve the usefulness of his Platina Ball and Torabasa Minore specials significantly, giving him much better ability to keep away, mix up, and do combos.

Watson starts at Level 1 and can reach Level 3. However, every time Watson is knocked down, he will lose a level. The number of hits towards your next level up is not lost if you are hit or knocked down, but you can't build up any hits if you are already at level 3 obviously.

You can tell what level Watson is currently at by looking at the aura around his staff:
No Aura: Level 1
Blue Aura: Level 2
Red Aura: Level 3
Specials
236A/C - Platina Ball ("Bubble")
Bubble projectile. Generally useful for combos, keep away, pressure, or fishing for hits. Super useful at Level 2 or higher. A version is high, C version is low.(?)
Level 1: Pops immediately. Generally very worthless at this level as it can't be used for any of the things it's good for.
Level 2: Bubble stays in place for a time, then floats forward and downward. Very, very useful.
Level 3: You can have two Bubbles active at once! (One for each strength of the move)
236B/D - Torabasa Minore ("Trap")
Bear trap. Grabs the enemy and holds them in place for a time, during which you can hit them as much as you like. B version places the trap in front of the enemy, D version places it behind them.
Level 1: Trap will only jump vertically. Only hits a few times.
Level 2: Trap will make much larges jumps and will attack horizontally if the enemy is near. Hits a few more times.
Level 3: Trap will make very large jumps and hits a large number of times.
j.236B/D - Tiny Drop
Air diving attack. Can be made safe if done low enough, but can often be unsafe even on hit. Can be very useful as a crossup however, especially when used with the Trap.
D version is slower, stronger, and knocks down on hit, but is even less safe.
j.214B/D - Falling Drop
A spinning ball attack. Remains fairly stationary but moves a little bit, and hits a fair number of times. Useful in some combos, but generally unsafe on block. Can also crossup.
Maintains momentum from jump, so can do some funny things if used while coming down from a jump.

Supers

236236B/D - Torabasa Mayore
Jumbo sized trap. Does a significant amount of damage on hit. Extremely good for mixups, amazing for lockdown. Good to use after a knockdown in the corner or to force people to keep away from you. Only activates if the enemy is in a hittable state, won't activate if they are knocked down or blocking.
B version places it near you, D version places it forwards a bit.
236236P - Alsuhail Almuhlif
Short range blast attack, hits 11 times. Very fast startup (5F). Doesn't do a lot of damage, but can be useful for padding out some combos, chipping someone to death, or punishing certain things at a distance. Startup is not invincible at all.
Counts as a projectile in terms of invincibility, but not in terms of hit-pause. Meaning, hit-pause caused by the projectile is felt by Watson as well.
Heat Up

Meteor Shower

214P - Twinkle Dust
While Watson is in Heat Up, Watson can input this move 3 times to activate a special meteor attack super. This super is extremely strong, and good for mixups and resets as well as chip damage. Very important to Watson's game!

Watson's Platina Ball is also enhanced so that both bubbles will activate in a single usage of the move. His juggle potential is also drastically increased which is very important as well.

Enhanced Specials

236A/C - Platina Ball ("Bubble")
During Heat Up, both bubbles will activate at the same time. Useful for many combos or just creating a wall of coverage. Level 1 bubble will still pop instantly.
j.236B/D - Tiny Drop
Recovery is reduced, possible to combo off of if used as an overhead.
j.214B/D - Falling Drop
In Heat Up, this moves pops Watson upwards regardless of jump momentum. This actually removes a lot of usefulness from the move.
Combos

Basics

A B 2C ...

2A 2A 2A 2D
Gets you a knockdown. Best you can get with 2A.
6A, 2A 2D
Works any time, but kind of crappy.
6A, A B 2C ...
Works VS crouching only. Which is generally when it will hit anyway. Doesn't seem to work on Marco at all.
... 2C, j.D j.2B, j.C
... 2C, j.2B, j.236B |> 2C
Basic, reliable combos.
(Corner)
... 2C, 236C, j.2B, j.C
(Corner)
... 2C, 236C, j.2B, j.214D
Strongest basic combos in the corner. Needs Level 2. Use j.C if you need to build up to Level 3, or j.214 otherwise.
... 2C, j.D 214B(3) |> 2C, j.C or 236236P
Glitchy advanced combo, needs to hit in a very specific way, but doing so allows you to get 3 C-Normals in one combo.
... 2C, j.214B(4) |> 2C, j.214B(4) |> 2C or 236236P
Extended advanced combo. Can be done midscreen VS some characters, should work in the corner on everything. Timing changes radically between characters.
MUCH stronger than his other combos and gets you a free level up, very worth learning.
Near Corner: You'll want to neutral jump instead of forward jumping here.
Urs: Do the j.214B very quickly after 2C.

Situational

Falling j.D 214D
(Corner)
Falling j.D 214D, 2C or 236236P
Generally unpractical, but can be useful.
Lv3 Trap Hits, 6A, j.C |> A A B 2C ...
Strong combo off a trap.
Trap Hits, 236236D
Strong combo. Damage is somewhat erratic. Generally stronger than any of his other super combos.
Setup Lv3 Trap, Throw (Trap Hits) ...
Combo off throw. Really stupid.
(Knockdown)
Setup 236D, 66, 2B (Trap Hits), ...
Low mixup with trap. Works at mid screen even with Lv1 because you can push them into the trap.
Setup 236236B, Back Throw
Need to set up the trap about one Watson length away, then back throw. If spaced right, the trap will spring.

High Gatchi Drive

j.C, 2C, j.236B |> C or 236236P
[501/952]
Basic combo.
Heat Up, j.B(1) j.2B, ( j.B(2) j.2B )*3, j.236B |> 2C
[?]
Heat Up, j.C |> 236P, j.D |> j.C |> 2C, j.236B |> C
[885]
Fancy but not ideal combos.
(1 Near Corner)
Heat Up, Jump Back j.C |> j.2B j.236B |> 4, 2C 236P, 236K, 5C 214P, 214P, 5A 5B 2C, 5C 236P, 214P, 236K or 236236P
[~1800/2200]
(1 Back to Corner)
Heat Up, 4, Cross Under 2C, j.236B |> 2C 236P ...
[~1800/2200]
(1 Mid Screen)
Heat Up, 6, j.C |> 236P, 66 6, 5C 236D, 66 66 2C 214P, 214P, 66 5A 5A 5B 2C, 6 5C 236P, 214P, 236K or 236236P
[~1800/2200]
Combo into meteors. Damage is somewhat random. Setup into the combo will change depending on the distance from the corner.

Low Gatchi Drive

(0 any)
66 j.C, B C 2C, j.D 214B(3), 2C, j.C or 236236P
[764/1269]
Another glitchy combo with 214B. Failing this, just use any normal juggle combo off 2C.
(1+ corner)
B C 2C 236P, 236K, C 214P, 214P, A A B C 2C 236P, 214P, 236K (236236P)
[~1800/~2200]
Very strong combo into meteors. Works on Consumer version.

Heat Up - Consumer

j.236B |> 5A 5B 2C 236P, 236K, 5C 214P, 214P, 5A 5B 2C, 5C 236P, 214P
Reset into meteors. They can escape by parrying the bubble.

Heat Up - Original

(Corner)
Lv3 Trap Hits, Heat Up, (2C, j.C) |> A A B 2C, 236P, 236K, 5C (Trap Hits) ...
In Original Mode you can loop this infinitely. On Consumer mode, they will be able to tech out after a launcher from the second bear trap.
(Corner)
Lv3 Trap Hits, Heat Up, (2C, j.C) |> A A B 2C, 236P, 236K, 5C (Trap Hits) 214P, 214P, A A B 2C, 5C 236P, 214P, 5A (reset) ...
Reset into Meteors. Lots of variations on this. Again, doesn't work in Consumer mode, they will always tech after the second trap.