These words are often confused. They sound very similar but they have very different meanings.
- To hold something as true. Example: I accept she may have been tired, but that's still no excuse.
- To receive something willingly. Example: I accept this award on behalf of the whole cast.
- To answer yes (especially to an invitation). Example: The minister would love to accept the invitation to your ball, but she has a prior engagement.
To express your willingness to do something, you have to use agree with a to-infinitive.
Accept is not possible in this case.
- She agreed to come. (= She expressed her willingness to come.) (NOT She accepted to come.)
- They agreed to part ways amicably. (NOT They accepted to part ways amicably.)
- My sister has agreed to look after my babies when I am away.