Answer:
   
   The Shulkhan Arukh (Orach Chaiim 343) provides a general guideline for
   raising children in an obervant home: As soon as a child is educable,
   the parents should teach the child about observance. Thus, even a
   three-year old (who, as a halakhic minor is not obligated to observe
   Shabbat) should be taught the relevant rudiments of Shabbat
   observance. This does not mean that the parent must take away a rattle
   or battery-powered toy, but it does mean that, on Shabbat, the child
   should be encouraged to play Shabbat-appropriate games. The R'ma and
   the Mishna Brurah (loc. cit.) point out that, by the time a child
   understands what Shabbat is (e.g., certainly by age 8), the child
   should be avoiding blatant Shabbat violation. The Shulkan Arukh's
   standard thus does not see the age of majority (13 for boys, 12 for
   girls) as a threshold for observance, and does not permit Shabbat
   violation by "educable" children (i.e., children above the age of 8 or
   so).
   
   In addition to the standard 39 classes of activities ("m'lakhot")
   forbidden on Shabbat, there is a prohibition against activities that
   are inconsistent with the spirit of Shabbat. The Ramban, for example,
   suggests that the obligation to "rest" ("shvita") on Shabbat is, in
   fact, biblical; it is in any event certainly a major rabbinic
   obligation.
   
   Thus, there are two general principles: Begin with Shabbat-appropriate
   games and play as early as is practical, and encourage Shabbat-
   appropriate activity, rather than Shabbat-inappropriate activity (even
   if such activity does not inherently violate Shabbat).
   
   What kinds of games and play are Shabbat-appropriate? Certainly
   activities with substantial Jewish content (e.g., board games with
   Jewish themes, available from many Jewish bookstores). Family
   activities should also be encouraged; reading stories, reviewing
   relevant parts of the weekly Torah portion, etc. Friday night bed-time
   can become a special occasion for hearing stories of "when Grandpa was
   young," or Chelm stories, or stories about SuperJew. When done
   appropriately, children see Shabbat as a special treat, not as a day
   when "we don't do these things."
   
   Some children's games [e.g., those involving explicit violations of
   halakha, such as games involving writing] are clearly inappropriate
   for Shabbat in an observant Orthodox or Conservative family. There is
   nothing wrong with saying "No, we do not paint on Shabbat." But it is
   educationally a much sounder practice to say "We don't ride bikes on
   Shabbat, but we do hear stories about Curious George going to shul."
   
   In some cases, the question of whether a particular activity is
   permissible on Shabbat requires halakhic expertise, and a rabbi should
   be consulted. For example, Conservative authorities permit swimming on
   Shabbat; most Orthodox authorities do not.
   
   Note that some Orthodox authorities rule that ball-playing is
   technically allowed within an eruv, based on the OH 306:45 and the
   Rama's gloss and the Mishneh Berurah's note on this. However, this is
   widely discouraged by rabbis as not being in the spirit of Shabbat.
   
   Rabbi Neuwirth's Shmirat Shabbat k'Hilkhata states that playing ball
   on Shabbat is okay for children, provided that it is within the eruv
   and on artificial surfaces. The reason for this is that Orthodox
   authorities feel that compacting dirt is a violation of a melakha
   (forbidden Shabbat activity); thus the restriction to hard surfaces.
   However, this should be checked with a local rabbi to determine
   whether it is appropriate for your particular community; don't assume
   beforehand that it is.
   
   For those that follow Conservative practice, in "A Guide to Jewish
   Religious Practice", Rabbi Issac Klein rules that some ball playing is
   allowed on Shabbat, based on the Rama on OH 308:45, as long as we
   distinguish between commercialized sports and activities one indulges
   in for personal enjoyment. Commercialized sports and amusements are
   obviously not reccomended because of the many violations of the
   Sabbath that are involved. Individual sports and amusements in
   themselves, where no other violation of the Sabbath is involved, are
   permissible. Again, a key aspect is that one should avoid
   participating in such activities to the point of overexertion and
   fatigue, which would make the act not in the spirit of the Sabbath."
   
   Hence, as a general principle, the best policy would be to give
   children lots of experience with Shabbat as a day for enjoyable
   Shabbat-appropriate activities, either synagogue- or family-centered.
   The emphasis should be on "shabbat-appropriate" activities.