What would be the most efficient game mode to grind for IP?

Whichever mode you're able to win the most frequently, though Co-op vs. AI doesn't really count (you get reduced IP for playing that mode as a higher level account). Riot has done a pretty good job of making the IP/game time pretty much the same across all modes.

I'm guessing bot games, the odds you will win is vastly superior to a pvp game and I recall the ip it gives is more or less the same

Short games like Dominion or ARAM will give you slightly more IP per hour.

You get a flat amount per win/loss plus a certain amount per minute spent in game. So 6 20-minute games will give you a little more IP than 3 40-minute ones.

Beginner co-op is the easiest way to guarantee getting first win of the day quickly.