Do I get priority after my opponent casts a Planeswalker? Say he casts a Tezzeret the Seeker. Can I kill him with a Psionic Blast before he can use any abilities?
No. The active player gets priority after a spell resolves. Barring shennanigans, it's going to be your opponents turn when the planeswalker resolves. They get priority, and can activate one of the abilities before you can blast it.
Also note that adding loyalty counters is a cost, so if your opponent activates Tezzeret's first ability, he will have 5 counters by the time you have the opportunity to cast Psionic Blast.
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Back in my day, Wrath of God didn't kill Black Knight and that's the way we liked it. We also had to pay mana for our 1/1s. Kids and your Memnites. Get off my lawn!
It's a very rare situation, but every once in a while you could conceivably have a Planeswalker enter the battlefield with another effect on the stack.
Edit: Or have an effect trigger on the Planeswalker entering the battlefield, which creates the same sort of situation.
In that case, since the stack isn't empty your opponent can't use his Planeswalker's ability yet, and after he passes priority you'll have a chance to cast your Psionic Blast before whatever else is on the stack resolves.
Also note that adding loyalty counters is a cost, so if your opponent activates Tezzeret's first ability, he will have 5 counters by the time you have the opportunity to cast Psionic Blast.
Edit: Or have an effect trigger on the Planeswalker entering the battlefield, which creates the same sort of situation.
In that case, since the stack isn't empty your opponent can't use his Planeswalker's ability yet, and after he passes priority you'll have a chance to cast your Psionic Blast before whatever else is on the stack resolves.