In english, you will find many situations where someone will ask you to do something you cannot, or do not want to do. Here are some ways to say that:
1) I'm afraid I can't..../すみませんが。。。(時間がないとか出来ないとか
Going to the bar with you sounds fun, but I have a class tomorrow, so I am afraid I can't go
I would like to see that movie with you, but I already made plans with my friend, so I am afraid I can't go.
2) I'd rather not/ すみませんが(何か)好きじゃないけど
I don't really like beer, so I would rather not drink that tonight.
There is someone I really don't like at that party, so I'd rather not go there.
3) Thank you for asking.....but I have to...../ 誘うて、ありがとうございますが。。。(バイトがあるとか他の予定があるとか。。)
Thank you for asking me to go camping with you, but I have to work tomorrow at my job.
Thank you for inviting me to your weedding, but I have to go to visit my friend in the hospital on that day.
4) I don't particularly like/ あまり好きじゃありません
Thanks for offering to share your nattou with me, but I don't particularly like smelly beans.
I have to listen to family when they tell me to do something, but I don't particularly like doing that.
Japanese and American culture are very different when saying 「no」so don't forget that is totally ok to say no in the US, and nobody will be angry at you. Just find a nice way to say it. ^_^