Since we've just learned about the ます form, here's a quick lesson on how to use it as an invitation.
Recall that adding か to the end of a statement makes it a question. By adding か to a negative, present verb, you create an invitation.
ばんごはんをたべません。 (bangohan wo tabemasen)
I won't eat dinner.
ばんごはんをたべませんか。 (bangohan wo tabemasen ka)
Won't you eat dinner (with me)?
Using the above statement by itself, it is implying that you are asking the person to eat dinner with you, or to join you for dinner. If you want to specify it as together, in the sense that it is only the two of you, you can use いっしょに (issho ni = together) or わたしと (watashi to = with me):
いっしょにえいがをみませんか。 (issho ni eiga wo mimasen ka)
Won't you watch a movie (together with me)?
わたしとえいがをみませんか。 (watashi to eiga wo mimasen ka)
Won't you watch a movie (with me)?
You can even use it to ask someone out and make it official between the two of you.
(わたしと) つきあいませんか。 (watashi to tsukiaimasen ka)
Won't you go out with/date me?
(The verb つきあう tsukiau means "to date." In its ます form, it is つきあいます tsukiaimasu)
That's all there is to this lesson. Now you can invite people to do things with you.