3 years old is definitely not too early to start playing structured games.

At first, early on, games should be very simple. The idea is to focus solely on the basic structure of games:

     --game start 
     --keep attention on game
     --turn taking/following in correct sequence
     --problem solving around a very simple goal
     --game end
     --with finally very clear assessment of player ranking (i.e. You won, I/we lost or I won, You lost)

Underlying all this should be a heavy focus on the maintenance of appropriate emotional responses and behavior modeling (e.g., no gloating, no sore losers, no emotional aggression or defensiveness).

Again, this is all ground-floor stuff, so kids can start learning it as early as 2-ish. Learning to take turns is a huge emotional advantage. As is learning how to both succeed and fail in games and life with grace.

A very good game that has a degree of complexity appropriate to kids that young is "Animal Upon Animal" by HABA.[0]

It's a simple stacking game: On your turn, stack this animal on that animal; if the stack of animals falls over, that's the fail state. (There is a 6-sided dice with pictures on 4 of its faces, and 1 and 2 pips on the the remaining, but it can easily be removed from the game until the child in question is ready for it.)

[0]http://a.co/bJ8syTL

Not at all. Candyland is recommended for children ages 3-6 and Chutes and Ladders 3-7. Both are good intros to the concept of board games.

I try to avoid telling a 3 year old what they are incapable of doing.

The 3 year old will tell you what they want to do.

At 3, my little guy's favorite toy was rated for 8+ years old, and his favorite game was chess.

He never cared if he won or lost at chess, his primary purpose was to push his pawns, castle, and then get in at least one en passant.

Yeah fair enough, I was just wondering about the future. How did you build up to a full board game with your kid? What were the phases? 1. Succeed in not eating the dice 2. ??? 3. Play a game of Sorry! within the rules.

The intermediate step here is what Candyland does.

It won't be very long and your child will be teaching YOU games that she/he learned from friends. :)