I'd use "that" for both.

In the former, it'd be we who, but you've added "are a group," and the ensuing clause is restrictive, so you need "that" to refer back to "group"...

In the latter, the ensuing clause is also restrictive.

We is the noun; what follows should agree with that. Though, using that sounds perfectly fine, too, if you are emphasizing the group, as opposed to the "we."

Variety implies choice, so I would go with which, in the second one.

I knew that one day I would find a case where "We is" is a proper conjugation of the verb "to be."

Just to be different, I would say "We have a variety of programs including yada, yada...."

And "The group does such and such." Or "This is a group that does...."

I think that when you leave off "We are", then you have the constuction of "A group that does....."I think that group would be the noun in the sentence.

I am a fan of James Kilpatrick. His viewpoint on sentence constuction is that the words should fall easily upon the ear. I think that he recently had a column about the use of who, which and that. I don't remember which he preferred. I think it was a case of which word would make the sentence flow more smoothly.

Bernstein says which is more common in the written word and that in spoken stencences.
And after "group" you use which, not who unless you say "group of people"...even though people is implied.
Not absolutely sure on this as I can't find my Fowler on English Useage...had a cleaning person who emptied and dusted my books and was creative about returning them to the proper shelves.
Linda C

The activities our group performs are...

We have a variety of programs including...

"that".

"that" also, but, I kind of like Don's wording. Smiles.