You know what they say, three's a crowd! Whether it's a dark kawa (coffee) on the Rynek (market square), a trip to the kino (cinema), a walk in the park (park; I know, no need for that one!), or a night on the town (impreza), it's always better have company. Let's take a look at how you go about inviting people to do things with you in Polish, suggesting activities and how to answer when people invite you.
Just as in English (where this is usually done by the use of possession or the first person - 'my party', 'I'll take you out’) there is a distinction in Polish between an invitation and a suggestion, or proposition. That is, one will imply that you are the one hosting, and therefore paying, while the other will simply suggest an activity that is to be done together.
To make an invitation that implies you are the one paying or hosting, you use the verb zapraszać (which literally means 'to invite') in the first person singular form, zapraszam (I invite), followed by the description of the activity you intend to do: Zapraszam [pana/panią] na herbatę (Let me take you out for tea).
Here the use of the pronoun pana for men and panią indicates a more formal indication, and the use of the accusative case for the associated noun (herbata -> herbatę), shows that it is the object of the transitive verb zapraszać (to invite).
Propositions and Offers
If you don't want to imply that you'll be the one paying, or if it's not something you are personally organising, there are a variety of ways to propose an activity.
You could ask if someone literally 'has (masz) fancy (ochota)' for something like a coffee or a tea, 'Masz ochotę herbatę?', making sure you use the correct accusative case endings for all of the associated nouns (ochotę and herbatę), the correct form of the verb 'to have' (masz [you have]), and inquisitive intonation to indicate that it's a question you are asking.
Alternatively, you can ask if someone is going to join you somewhere, using the dynamic też (also) and the relevant verb of movement. For example I could see if someone is joining me on holiday by asking them, jadę do Hiszpanii, jedziesz też? (I am going to Spain, are you going too?). Notice here the case conjugations for the object - Spain - and the correct use of the verb of movement (Jechać, which indicates travel by transport not by foot), along with altered personal endings that change the subject of the verb, from me to the person I am directing the question at.
Replying to Invitations and Propositions
To reply to an invitation in the affirmative you can, of course, simply use tak, but there are some much better ways of letting your partner know if you will be joining them or not, below is some useful vocabulary:
Oczywiście (Of course)
Niestety, nie (unfortunately, no)
No pewnie (This one's a little more casual and means somethign like, 'sure thing!')
Dłaczego nie? (Why not?)
Nie mogę (I can't)
Bardzo chętnie (I would be very glad to)
Nie wiem (I don't know)
Przepraszam, nie (I'm sorry, no)