Yes. You should.
You may or may not enjoy it, but you should at least give it a fair try - which probably means getting to about 50 wins to get past the initial shock.
Pet PvP is the most anonymous thing you can do in WoW. Your opponent sees your character as a random NPC, and no character names or realm are shown, so unless you name your pets something specific, nobody will ever know who you are - and even if you do name your pets, that's pretty hard to track down. Think of it as an instanced battle against a hidden - and cunningly malicious! - pet tamer.
I identify four stages of PvP battling:
1. Stage 1: WTF is happening?
Just the mechanics of PvP - being transported, instanced, sizing up the opposing team, choosing a pet to go first, and watching the timer ticking down - feel weird when you begin, so you need to write off your first attempts to learning these basic ropes.
2. Stage 2: Why did I just lose?
Unlike PvE, a lot of PvP involves trickery - buffs, swaps, weather changes, and especially head games. Head games like: you have an Iron Starlette against an Idol. When do you use your nuke? when does the Idol use its Deflection?
Especially, you need to size up the opposition's team in the first few seconds, and figure out his plan. If you can figure out his plan, you have a shot at disrupting it. If you don't, he gets to do what he wants.
YOU. WILL. LOSE. A lot. That's OK. PvE battling, past the initial experience at 25, hands you a 100% win rate without thinking - other people can do your thinking for you. Remember, for every win in PvP, there's a loss. Actually, there are more losses than wins, since Draws are effectively losses in PvP. Losing is not a problem. Just figure out why you lost, what you could have done better. The unexamined battle is not worth engaging. The addon Pet Battle Log Keeper will preserve a log of each battle, so you can figure out what went wrong. It also may explain how your opponent went first, or swapped you out, when you don't recognise the effects of a move.
3. Stage 3: Meeting the Meta
At any given point, there is a set of favoured pets that is good for PvP - the "meta". Since I started in Warlords, I think it's fair to say that about 50 pets in specific breeds constitute 90%+ of what you will see, and these pets will be used in specific combinations to build synergy. Typical examples are AoE teams, buff teams, weather teams, Clonedance teams, control teams that swap you or themselves out to disrupt your plans.
You will start seeing these patterns again and again, and you will start to recognise them.
are views on the current meta.
You should also check out YouTube channels Rosqo, Llennoca, and Discodoggy to get team ideas and understand what's happening in a PvP battle.
In this phase, you likely be getting team ideas from others, and will be winning gairly regularly but still learning from other people. Especially, learn from the people who beat you!
4. Stage 4: Mastering the Meta
This is the phase where you have learned the tricks, and are confident enough in entering the queue that you are trying out your own original team ideas and making teams that are deliberately weaker than you could choose, for fun, or to avoid getting everyone in the queue that day all riled up.
So I've talked about what to expect. I haven't said WHY you should try PvP. You should try it long enough to see whether you like it. Not many people do really like it, but those who do have found another great activity to enjoy in WoW, and that's worth a lot. Though I have to say that PvP rewards suck when compared with PvE, you can also get achievements, Stunted Direhorn, and a title from it,
This thread also talks about starting PvP
and is well worth reading, though my description of the three dominant meta strats has changed. Clonedance and Darkness were been nerfed at the start of BfA, and Haunt now has a big penalty, so is hard to justify in PvP. You will still see these strats, but they are not the overwhelming majority they used to be - well done, devs!