Thank you for trusting us with an answer! You've got a dilemma and we truly feel for you, because most of today's kids are not used to obedience to YHWH. Unfortunately, our world centers around weekend games and activities for the kids. But the bottom line is: What and WHO is more important: Obeying YHWH or making the kids and their coaches happy? It's a very "ticklish" situation, especially if you've just started becoming Torah observant and the kids are used to doing their "Saturday sports" etc.
Anyway, the short answer to your question about whether sports and traveling on the Sabbath is okay is: No.
Exodus 20: 8 "Remember the day, Shabbat, to set it apart for God. 9 You have six days to labor and do all your work, 10 but the seventh day is a Shabbat for ADONAI your God. On it, you are not to do any kind of work -not you, your son or your daughter, not your male or female slave, not your livestock, and not the foreigner staying with you inside the gates to your property. 11 For in six days, ADONAI made heaven and earth, the sea and everything in them; but on the seventh day he rested. This is why ADONAI blessed the day, Shabbat, and separated it for himself.
Isaiah 58: 13 If you hold back your foot on Shabbat from pursuing your own interests on my holy day; if you call Shabbat a delight, ADONAI'S holy day, worth honoring; then honor it by not doing your usual things or pursuing your interests or speaking about them. If you do, you will find delight in ADONAI - I will make you ride on the heights of the land and feed you with the heritage of your ancestor Ya'akov, for the mouth of ADONAI has spoken."
The longer answer is: How much of their day is going to be spent traveling and participating in their games? If it's just a couple hours or so away from home, then that wouldn't be too bad.
However, the Sabbath is supposed to be about YHWH and studying His Word and RESTING, and "family time" in the process of resting from our weekly work and routines. Are we "resting" if we're carting the kids around to get to their various events? Are the kids "resting"? Are their minds on YHWH on Shabbat if they're concentrating on their games and winning, etc., especially if the travel itself takes hours? And let's not forget, if you travel a long way from home, you end up buying gas and eating at a restaurant, thereby "causing others to work." Part of the Shabbat rules include not committing commerce on the Sabbath.
Nehemiah 10:31 "If the peoples of the lands bring merchandise or food to sell on Shabbat, we will not buy from them on Shabbat or on a holy day.
Certainly, it's hard to keep very small children from playing games and doing what small children do - and YHWH never said we had to keep our kids from playing games and just "being kids." And playing with your children on Shabbat isn't considered "work" either, because part of family time means "having fun" and just being together. But when we have to drive our children miles away or even to another state to compete in competitive sports, that gets a little stickier because, suddenly, the entire Shabbat is taken up with what we want to do....
Of course YHWH doesn't expect us to spend 24 hours in the Bible on Shabbat, either. Realistically, not many people could spend all day long in the Bible because our minds need an occasional break. Too much reading and studying ultimately ends up being "work," which would defeat the purpose of "resting" on Shabbat. But He does expect us to remember all day long that it IS Shabbat and that we ARE to "rest" and take a break from "the world." It's all just a matter of common sense.
Deuteronomy 6:4 "Sh'ma, Yisra'el! ADONAI Eloheinu, ADONAI echad [Hear, Isra'el! ADONAI our God, ADONAI is one]; 5 and you are to love ADONAI your God with all your heart, all your being and all your resources. 6 These words, which I am ordering you today, are to be on your heart; 7 and you are to teach them carefully to your children. You are to talk about them when you sit at home, when you are traveling on the road, when you lie down and when you get up. 8 Tie them on your hand as a sign, put them at the front of a headband around your forehead, 9 and write them on the door-frames of your house and on your gates.
So yes, by all means, allow the kids to play and even watch PG rated movies or to play computer games for a time - but NOT for the entire day or hours on end. There HAS to be "YHWH time" on Shabbat where you do a family Bible study and teach your children the importance of obedience to God. They need to learn that HE is the center of Shabbat and that the day is all about resting from their normal activities - school, competitive sports and whatever else they do during the "work week." You'll know how to gauge how much your kids can understand and are willing to "give up" on Shabbat. Some things are simply not allowed on Shabbat - and those things are activities that detract from keeping the Shabbat "holy."