You'd be better off playing on valve official servers once you're more confident with your play, but as a starting point, why not just play offline with bots? You won't have to worry about ping as you'll be playing locally.
Playing offline, you also don't have to worry about team mates, you can simply practice your aim and shooting with the various guns on bots. Bots for most players with a bit of experience are easy to play against, even on the expert bot level.
I would recommend playing offline with bots of the lowest level first and work your way up to expert bots as it becomes easier. Once you find playing against them easy, move on to official servers and play against real players in the various modes.
The casual servers have a wide range of skill groups, so you may feel that you're not ready, but with things like deathmatch you are playing on your own to maximise your kills, so again you don't have to worry about letting your team down. This is also a good place to warm up as stated in the other answer.
Also, there a lots of community made warm up / aim maps that a lot of players use before matches.
Fast Aim/Reflex Training Map [SP vs Bot Map] / aimtraindriving
Once you're a bit more confident, I'd suggest competitive mode would best suit you as you'll play with people in a similar skill group to you. You do have to commit to playing the match though, which usually averages any time between 30-60 minutes at the lower skill groups.
To play competitive mode you need to obtain Level 3 - Private Rank in their XP system, which can be gained by playing on official servers.
With official server, you're less likely to get team killed. In casual mode, team damage / friendly fire is off and in competitive mode, you get punished for repeatedly killing team mates and you can be reported for griefing.