If you need something to play with during service, I would guess that you are either bored or lost. I recommend you try one of following:
(1) Sign up for a class in Jewish liturgy. Contrary to popular belief, one does not learn to pray by osmosis or intution alone. Jewish prayer requires a good deal of knowledge, and I know of no one who can decipher the siddur without plenty of help.
(2) Ask a rabbi for a recommendation of a book or two to help you learn about Jewish liturgy and about prayer. Start with one prayer and learn enough about it to make it meaningful to you.
(3) Try learning some new Hebrew vocabulary: study the siddur.
(4) Try using the prayers as starting points for meditation, if you are a meditator.
Here are some books about Jewish liturgy which may be helpful to you:
• The Enchantments of Judaism: Rites of Transformation From Birth Through Death by Jacob Neusner
• Jewish Liturgy by Esmar Elbogan (trans. by Raymond P. Scheindlin)
• Jewish Liturgy and Its Development by A.Z. Idelsohn
• Jewish Worship by Abraham Millgram
• Hasidic Prayer by Louis Jacobs
• To Pray As a Jew by Hayim Halevy Donin