We've had really good success with Magic Maze Kids - as well as Ticket to Ride: First Journey and Catan Junior (both of which are fantastic). The thing that Magic Maze Kids has going for it is an _absolutely fabulous_ tutorial system that slowly introduces every mechanic of the game.
Other games like Pigasus and Snowman Dice (both by Brain Games I think) have been a hit too.
I think if you're willing to be VERY flexible with all rules, 2.5 is old enough to play all of the above.
When we play with younger kids we try really hard to emphasize fun and downplay stringent rule following, for better or for worse.
Some games, like snowman dice, are fun just for the sake of it. It's fun to roll the dice and push them around.
I'd say Catan Junior is fine for a 3 year old, but make sure to spend the first few times just familiarizing the rules (i.e. making the connection between getting a resource and spending it by matching up the resources on your resource card).
Ticket to Ride Junior is a little bit harder, but a 3 year old should do ok with help. Again, if you really "take it easy" about the rules a very young child can have a lot of fun with these if for nothing other than the colors and manipulatives/meeples alone.
Magic Maze Kids we did starting at 2.5-3ish and it was fine.
Pigasus, on the other hand, a 2 year old can do and really enjoy. That game is absolutely amazing because it's one of those things that a small child can do better than some adults (like... for real!). Also the sound the pig makes when you squeeze it is incredibly funny to most kids and very rewarding. It's a fun one.
My 3yo daughter can handle Ticket to Ride First Journey, and even now knows some of the city names (we actually have a German edition, since I couldn't find an English one at a sensible price). It plays a bit quicker than the full version, so tends to hold her attention longer.
Other games like Pigasus and Snowman Dice (both by Brain Games I think) have been a hit too.