My brother's dog chewed/scratched my door frame in our apartment and I was looking for tips on how to fix this ourselves without hiring someone before we decide to move out, the apartment is pretty old so I was not sure just going to a home store would do us any good... Thank you!
It looks like the casing sustained most of the damage. It appears to be aged pine or fir. The challenge will be finding a matching profile. You may need to replace the casing on the entire opening face so that the three pieces match.
Look for a molding with the same width as the original and a profile of similar style. Depth isn't critical as long as it's aesthetically similar. A miter saw and small trim nails will be needed. Chances are a couple coats of polyurethane varnish will give a similar look after some time for UV light to darken the pine.