Learn Russian: Давайте пригласим их в поход.

The ending -те shows that we are addressing either multiple people or someone we speak to formally (using вы). If you were suggesting this to just one close friend, you would drop the ending and say Давай пригласим.

Why is пригласим conjugated instead of being in the infinitive?

When making a "Let's..." suggestion with a perfective verb (a completed, one-time action), you must use the "we" form of the future tense after Давай or Давайте. Since пригласить is perfective, it becomes пригласим. If you were using an imperfective verb, you would use the infinitive instead.

We say "on a hike" in English, so why does Russian use в instead of на?

In Russian, a поход (a hike or camping trip) is treated as an activity you go into, so it takes the preposition в. We are implying movement or destination here, so поход is in the accusative case.

Why doesn't the word поход change its ending in the accusative case?

Because поход is a masculine inanimate noun (it is a thing, not a living person or animal). Masculine inanimate nouns keep their dictionary form in the accusative case.