Level 20 isnt as long a grind as you think. At level 10 you get a stimpack that gives you +100% experience. So you will end up with around 60000-80000 xps for a match, then you get that amount again for the stimpack bonus, and then you can get another 50000 on top of that if you play with someone from your friends list. So you can average about 190k a match.
The only problem you might run into, is playing in Practice and Versus AI mode has an experience cap every 24 hours of 2.25 million xps. After you cap out, you have to wait til 24 hours after the first match was applied to the cap. So if you started playing at 6am and capped out later in the day, your cap will start releasing xps at 6am the next day. This cap is not applied to Quick Match or Hero League, but those are Player against Player matches.
You will find playing with 4 other players results in more wins than you playing with 4 AI on your team, as the AI on your team plays like boneheads, while 4 other players might know what they are doing and help you get a win (easier).
* Dont die. Run before you die or you give xps to the other team and make their team level faster. If you get low on mana and/or life, use your hearthstone. You will appear back in your base in the shielded area and regen life and mana at a much faster rate.
* Make sure there is at least 1 person on your team in each lane prior to level 10. Your team gains xps when minions die or buildings are destroyed. If no player is nearby you dont gain those xps. So you want to have at least 1 player near those non-stop minion skirmishes that happen midfield early on to "soak" xps for the team.
* Winning the objectives are huge for your team, dont ignore them. Even if you see some of your team there, go help. You could be what tips your scale in your teams favor.
* I have noticed other players are more tolerant (not always) if you let them know ahead of time you are new. You will get nasty people sometimes, but thats in any game. Dont let it get to you.
* When I first started playing, I tended to pick Passive talents as it made the game "easier" for me. But as I learned the game I realized that those passives are alot weaker than the other choices in most cases. I started looking at builds at this website: http://www.heroesnexus.com/heroes
and quickly learned which talents were better choices. Something to point out about that site, is most of the builds are for Player vs Player, so they choose the levels 13, 17 and 20 talents based on that. Once you get the hang of the game you will find that those listed builds can be tinkered with and changed to help more in an AI game as the computer reacts in a patterned way.