You might like to check out two responses I've made to this question, using a different account. There is always more to be said though, so feel free to ask if there's anything it does not cover.

The point of the Allied intervention was to restore an Eastern Front against the Germans, although Japan's participation was marked by naked expansionism in Siberia. Greeks and Serbian troops fought as part of the French-led Armée d'Orient. After the armistice with Germany in November 1918 the political objectives of the intervention became ill-defined and it quickly grew unpopular both at home and with troops on the ground in Russia. The French fleet in the Black Sea did mutiny, in fact.

This answer explains the motivations and gives an overview of the Allied Intervention, and also discusses the French participation more specifically.

This one looks specifically at America's role in the civil war.

For some further links, a couple of answers I've written: this one talks about interventions in the Russian Civil War generally, and this one details US involvement.