From Final Fantasy - Wiki on Neoseeker

Blitzball Techniques in Final Fantasy X] can be learned through "Techcopy". By marking players on the opposing team, there is a chance your player will learn the ability should that opponent use the ability during the match. When they do use it you will be able to learn it by pressing X at the correct time. During the time that they carry out the technique a little flashing "Techcopy" sign will come on the bottom left hand corner of the screen, this will allow you to copy the technique, its said that you have to press X when its a certain colour, however after much experimenting it can be said that you have to press it during a certain part of the animation. The part varies for each technique as does the animation, however here's a full techlist followed by some tips on when to press the X button when trying to learn certain techs:

First Page
Jecht Shot	Wither Shot	Wither Pass 2
Jecht Shot 2	Wither Shot 2	Wither Pass 3
Sphere Shot	Wither Shot 3	Volley Shot
Invisible Shot	Venom Pass	Volley Shot 2
Venom Shot	Venom Pass 2	Volley Shot 3
Venom Shot 2	Venom Pass 3	Venom Tackle
Venom Shot 3	Nap Pass	Venom Tackle 2
Nap Shot	Nap Pass 2	Venom Tackle 3
Nap Shot 2	Nap Pass 3	Nap Tackle
Nap Shot 3	Wither Pass	Nap Tackle 2
		Next Page
Key Techniques: ..........................................
Second Page
Nap Tackle 3	Anti-Venom 2	Brawler
Wither Tackle	Anti-Nap	Pile Venom
Wither Tackle 2	Anti-Nap 2	Pile Wither
Wither Tackle 3	Anti-Wither	Regen
Drain Tackle	Anti-Wither 2	Good Morning!
Drain Tackle 2	Anti-Drain	Hi-Risk
Drain Tackle 3	Anti-Drain 2	Golden Arm
Tackle Slip	Spin Ball	Gamble
Tackle Slip 2	Grip Gloves	Super Goalie
Anti-Venom	Elite Defence	Aurochs Spirit
		Previous Page
Key Techniques: ..........................................

Anti-Techs - For any anti-techs (level 1 or level 2 such as Anti-Venom and Anti Drain) you should press X just as the Tech copy bit is about to disappear
Drain Tackles - For all drain tackles the techcopy bit is only there for approximately 1 second so as soon as you see it press X
Grip Gloves - You should press X when the techcopy sign is white
Invisible Shot - Press it JUST as soon as the ball is released, it may be easier for you to win this in a tournament as it requires impeccable timing
Sphere Shot - You must press X just as the ball is kicked, but be careful not to do it before as it wont work unless you press it AFTER the ball has made contact with the players shoes

Experience

Players are awarded experience points for every move they successfully perform:

Action	Ordinary Action	Lvl 1	Lvl 2	Lvl 3	Successful Execution Bonus
Nap Shots	1	2	3	4	1
Venom Shots	1	2	3	4	1
Wither Shots	1	2	3	4	1
Jecht Shot	1	3	6	-	1
Aurochs Spirit	1	6	-	-	1
Invisible Shot	1	3	-	-	1
Sphere Shot	1	2	-	-	1
Volley Shot	-	3	3	3	1
Nap Passes	1	2	3	4	1
Venom Passes	1	2	3	4	1
Wither Passes	1	2	3	4	1
Nap Tackles	0	1	2	3	2
Venom Tackles	0	1	2	3	2
Wither Tackles	0	1	2	3	2
Drain Tackles	0	1	2	3	2
Block	0	-	-	-	3
Catch (Keeper)	-	-	-	-	4
Deflection	2	-	-	-	-
Super Goalie	-	2	-	-	-
Tackle Slip	-	1	1	-	-
Anti-Tech	-	1	1	-
Good Morning	-	4	-	-	-
Gamble	-	4	-	-	-

From the table you can clearly see that shots are a great way to earn EXP especially the special ones that Wakka and Tidus can learn. The best trick to do is pass between your characters during the first half as this will increase their levels for sure (they will get around 120 points if using a level 3 pass).