I played sports growing up and learned many valuable lessons while on the field. I want my daughter to learn some of those same lessons. Is it wrong of me if at a young age I start to push sports on her? Enroll her in soccer, t-ball etc.
I think it's important to push your kids. I also think it's important not to push too hard. Balance in all things.
Encourage her. Play with her yourself and make sure she enjoys it. When she agrees, then sign her up. Make sure she understands that she'll have to stay with it for at least some period of time -- it's a lot more fun once you've gotten past the initial awkwardness.
No. You should not push any recreational activity, no matter how rewarding you found it, on her.
However... it is perfectly appropriate to encourage her to try sports.
The difference between pushing and encouraging is kind of subjective and hazy, but my interpretation is that "pushing" occurs when the child has already formed their opinion, and you enforce behavior that is contrary to that opinion.
If you have your daughter try football, and she doesn't like football, continuing to pressure her into participating in football is pushing, and is not likely to result in a life-long passion for football.
If she doesn't like football, then you should encourage her to try something else, until she settles on one or more sports she does like.
Depending on the age and personality, it may be appropriate to make it a requirement for the child to give a sport that initial attempt. Much like a policy of trying everything on your plate at dinner, enforcing "you must try a sport before deciding you don't want to do it" is perfectly reasonable.
What "try" means is something you should be very clear on ahead of time (i.e. is it one session? One week? A full season?).
It is also reasonable to establish an expectation that, while the child may have veto power over individual sport activities, they have to pick something, particularly if you are focused on the health and educational (teamwork, goals, strategies, etc.) benefits of sports. Just be wary of budding lawyers... make sure that you address ahead of time whether you consider "NASCAR" a legitimate sport for your child to pursue, for example!
Let me tell you a story: My ex-wife's father, a military career person, forced her brother into riding bicycles from early childhood. The boy hated it, sucked at it, but was forced into doing it anyway. One of those times resulted in a nasty fall, a broken leg, and a broken relationship between him and his father that never healed. That boy turned into a man and never spoke a word with his father. His father's like dead to him.
Would you like to experience the same? Is it worth it?
While I totally agree with the idea that pushing or forcing your child to do extracurriculars they are not interested in in any way is wrong, I also think physical activity is super important too. My rule with my own daughter has been that she must be doing one activity that involves physical training. This encompasses a lot of things from the team sports you mention to individual sports such as many track and field events or even dance. As it stands now, she has discovered she really enjoys tennis and Taekwondo so that is what she does.
For some kids, team sports are fairly intimidating. If you are worried about exposure to lessons learned from "team" exposure, there are other ways for them to get exposure to these types of skills. Debate teams need to work together for example, as do many other groups. To address these types of skills, my daughter has chosen instead of a team sport, theater. She is now involved with a community theater which gives her a lot of exposure to some of the same things "teams" have to learn in team sports such as awareness of others on the stage, knowing how to share the spotlight (or the ball) and even strategy with other actors and actresses about how to work together to depict the emotions aimed for most clearly and accurately.
Anyway, this plan has worked to give her enough choice not to feel "pushed", but still meets my need of wanting her to have the exposure to the lessons to be learned from such activities as well as the physical exercise.
It has worked really well for us.