Learn Icelandic: Við lærum alltaf saman.

Because those specific endings only apply to singular pronouns (like þú, hann, and hún). When using við (we), almost every single verb in Icelandic ends in -um, completely hiding what group it belongs to! If you changed the subject to "he", you would see its Group 2 ending: hann lærir (he studies).

Does læra mean "to learn" or "to study"?

It covers both! Icelandic uses að læra for both the act of sitting down with your books ("studying") and the process of acquiring new skills or knowledge ("learning").

Why doesn't the vowel in lærum change because of the -um ending?

You might be remembering the "U-shift" rule from earlier lessons, where an a in a word changes to ö if the ending contains a u (for example, tala becomes við tölum). However, the vowel æ is immune to this rule, so it stays exactly as it is.

Why is alltaf placed after the verb instead of before it, like in English?