join (to become a member of, participate in): He decided to join the army. Will you join me on a walk? Would you like to join us for lunch?
take part (join in or be a part of an activity): How many competitors will take part in this race?
participate rather formal (join in or be a part of an activity): They have agreed to participate in literacy classes.
engage in something formal (to take part in a particular type of activity): You are free to engage in peaceful political activity.
enter rather formal (to start to work or become a member of): He entered the army at the age of 19.
join in (to take part in something with others): Will you join in with us to clean up the hall?
get involved (take part): I don’t want to get involved in such an argument.
have a hand in (play a part, have a role in, take part): I want to have a hand in this decision.