I think getting up pet resistances now is pretty good, better than ever even with a lot of changes. There are 4 main avenues for acquiring them that range from just starting out all the the way to end game:
Mastery Skills. Occultist gets Storm Spirit and Aspect of the Guardian to provide pets with free Elemental, Acid and Physical resistances. Shaman gets Oakskin and Emboldening Presence to provide pets with free Pierce, Aether, Bleeding and Physical resistance.
Necromancer gets it the worst just getting Master of Death to provide pets with Vitality resistances but Occultist/Shaman having the dearth they have is already an amazing start and taking even just one mastery out of them (e.g. Cabalist/Ritualist) is already a substantial boost.
Devotion. Various constellations are amazing sources for them. From experience, Wolverine is bountiful and immediately accessible. It’s affinity like Shepherd’s Crook also aligns with Mogdrogen. Typhos is in a similar position as well.
Equipment. A lot of Epic and Legendary equipment feature resistances for both the player and pets by end game. Before then, faction equipment is also notable for doing the same and with relevant pet stat bonuses and other nice things (like % pet damage/OA). For example:
Rhowari Shoulderguards.
Rhowari Cuirass.
Bysmiel Stormshroud Cuirass.
Solael Vile Pauldrons.
Coven Tamer’s Hood.
Coven Lifebinder Vestments.
Components and Augments. End game has Seal of Might which is a well-known and used one. Before then, there is Purified Salt and Imbued Silver for a chunk of pet Aether/Chaos resistance whilst also providing Aether/Chaos (and Bleed in the case of Imbued Silver) resistance to the player.
There are various augments for pet resistances, some of which are available mid-game at Honoured reputation for Devil’s Crossing, Malmouth Resistance and Order of Death’s Vigil:
Rifthound Salts.
Mender’s Powder.
Binding Dust.
More options open up at Level 90 for Acid and Bleeding resistance but I wanted to highlight a few again that can be used while levelling that aren’t too hard to get ahold of. Buying some of these and rotating around them to fill in gaps that the above methods don’t cover is a good idea. It’s also not a bad idea to use the one that covers whatever resistance you need the most for any given current area - Mender’s Powder is especially good for Homestead up to the Loghorrean and for the Barrowholm Ch’thonic rift up to Theodin as examples.