As Christians we want to serve God is all the ways He wants us to. Does Judaism call on us to observe the Shemitah year by obstaining from our farming and gardening? What about cutting fire wood for heat? Is there anything else shmita-related that we should do? Or is this just for the Jewish people who worship God?
I can't answer for Christianity's view on this, but here is the Jewish view: the shmitta year applies solely to Jewish farmers in the land of Israel. So, from a Jewish perspective, as a non-Jew, there is nothing shmitta related you are obligated to do. If you have a farm or garden, even if it is in the land of Israel, Jewish law places no restrictions on your farming with regards to shmitta.
I have lived on a kibbutz farm in Israel during shmitta, and via a Rabbinic strategy pioneered by Rav Kook called heter mechira (permission for selling), Jewish farmers sell their land to non-Jews for the year, because the produce of non-Jews is, according to this leniency, acceptable. (I believe there are some Haredi Jews who do not accept this heter, but I cannot confirm that at the moment) So, if you are in Eretz Yisrael during shmitta and want to be helpful, you could assist Jewish farmers by being a signatory to the legal transfer of the land for the year. (However, Jewish farmers generally don't have trouble selling their land temporarily).