1.
to start on; to begin working on
To begin doing something, especially a task or project that one has been putting off or hasn't addressed yet.
2.
to touch; to lay a hand on; to tamper with
To physically touch or handle something, often with the nuance of meddling with or tampering with something one shouldn't.
A common idiomatic expression using 手 (hand) and つける (to attach/apply). The two senses are connected: both involve putting one's hand to something, whether figuratively (starting work) or literally (touching/tampering).
Usage
Sense 1 is very common in everyday conversation, often in the negative (手をつけていない — haven't started yet). Sense 2 often implies doing something improper, such as spending savings one shouldn't or interfering with something.