Succeeding with Positional play...or am I?
you need to sharpen your tactical sense as this will help you win more than anything else at this level. it's okay to positionally suck in the meantime. that's not as important. in any case, tactics dominates the game between people who are not gms more often than not. however, if you don't like playing tactically then you are going to have a hard time going forward. you will later realize that no matter what style you intend to play in, you will need to be really good at tactics first.
3...a6 does not contribute to development or the center, but neither does 4 a4. Better 4 d4.
7 Bd2 is passive. What is the bishop doing there? Better 7 c3 to prepare d4, which you could have played on move 4.
20 Qxc8 is a blunder. God save the Queen! 20 Qb7.

Good thing you had a clear plan. But I agree with what has been said before.
You state that 3...a6 hands you the initiative. That's only true if you try to punish him/her by opening the centre / attacking. After playing 4.a4 and 5.d3 Black's pawn on a6 is perfectly fine.
7.Bd2 - The bishop looks bad there and doesn't do much except from blocking the knight. You need to play c3 anyway if you want to expand with b4.
Not sure if it is good but did you consider f4 on move 14 and 16?

Good thing you had a clear plan. But I agree with what has been said before.
You state that 3...a6 hands you the initiative. That's only true if you try to punish him/her by opening the centre / attacking. After playing 4.a4 and 5.d3 Black's pawn on a6 is perfectly fine.
7.Bd2 - The bishop looks bad there and doesn't do much except from blocking the knight. You need to play c3 anyway if you want to expand with b4.
Not sure if it is good but did you consider f4 on move 14 and 16?
thought you were blueemu for a minute
I was going to post some analysis, but I think the other commenters have covered all the main points very well. I would just like to make a couple of comments
3...a6 was an obvious waste of time. It's true that in the closed Italian Game, a6 is often played. But this isn't a closed Italian--yet! It was time to change plans and open the game up with a quick d4!. You determinedly forced the game back into quiet channels, where his a6 move was quite appropriate. But you didn't have to. When your opponent makes suspicious moves, don't just stick with the plan. You have to take your chances when they arise
7.Bd2 is a surprisingly poor move. If you were intending to play a quiet Italian, you should understand that you are probably going to play some combination of O-O, Re1, Nbd2, and Nf1. You bishop on d2 is not only fairly useless, it jams up your whole development
I used to play a lot of chess back in the 90s (2-time Manitoba Scholastic Champion, Grade 10 in 1992, Grade 11 in 1993), but looking back I see that was mostly from being part of a small talent pool, and I recognize how spotty my play was. Basically, I played to smash, and was usually able to out-calculate my opponents, so I won a lot, in spite of a lot of bad habits and a generally poor understanding of positional elements. Probably equivalent to about a 1600 rating, 1700 at my best.
Getting back into the game now after nearly 20 years of inactivity; my tactical ability is somewhat rusty, but my more mature self has a better grasp of what position means, so I've been trying to focus more on broad strategic considerations instead of just trying to "brute force" every move. In the game below, I (White) had a won position before I got distracted and blundered it away (ignore everything after Black's 18th move), but I'd like some input on how well my approach would have stood up against a stronger opponent. I can't seem to annotate the PGN properly, so I'll add my notes below.
Key moments that I'd like feedback on:
3...a6. Passive move by my opponent hands me the initiative.
4. a4. Blocking 4...b5 and playing for space on the Q-side. 4. d4 was a possibility, but I generally dislike opening the center early in e4 e5 games. Personal preference more than anything I guess.
7. Bd2. Supporting my planned advance of the a and b pawns. I'm a big fan of this B position, although 7. c3 would probably have done the same job better. Even after the played move, I still need to play c3 before b4.
7....Nd5. Black helps me advance the b-pawn by removing one of the defenders of b4.
8....exd5. Black imprisons his own DSB. I decide to make sure it stays that way.
9. b4 Bb6 10. a5 Ba7. Q-side expansion achieved.
11. Na3. My plan here was 12. Ba2 and 13. Nc4 to pressure the center and be able to take advantage if Black pushes the c and d pawns.
13. Ba2. B is far too powerful to trade off Black's N. Plus, I was planning that move anyway.
14...Nxc4. Black had dangerous attacking chances against my K-side, which he just threw away with this exchange.
15....f6. Blocks any shenanigans I may have planned against f7 after playing Qf3, but now Black can't 0-0
17. e5. Probably the strongest move I played all game. With Black's pieces lacking coordination, time to open some lines.
19. Qxc7. The first of several horrible moves. Ignore this move and anything that follows, as it has nothing to do with sound strategy or tactics. I had many ways of crushing at this point, and played none of them.
So to summarize my game, my goal was to claim space on the Q-side, restrict Black's pieces, and eventually open the e-file for my Rooks. Was my approach generally sound, or was I just lucky that my opponent basically let me do as I pleased for most of the game? Opinions and criticism welcome.