When we bet flop and pick up significant equity on the turn, I've heard 2 opposite thought processes.

1. We picked up equity, so we have a great opportunity to semi bluff.

2. We picked up equity so this is a great spot to check back and take a free card.

I generally first consider this: How likely is villain to check raise me?
But I dont really know how I should react to them check raising more/less. If they rarely check raise, I should probably bluff way more often right?

Ok so lets say villain check raises often. Should I be more likely to bet a gutshot that I turn, or a stronger bluff like a flush draw or open ender that I turn? Thx

The decision to continue doesn't always have to be either/or and in fact can be mixed between continuing and checking back. Some factors to consider:

How often does villain fold to cbets and barrels?

How does the turn card interact with my range vs my opponents' ranges?

How much has this card improved my equity or how much equity do I have now?

What will river play look like? Will I get a significant amount of villain's stack if my draw comes in? Will villain fold often to a 3rd barrel?

All of these questions can affect whether you bet or check with the same hand in the same scenario.

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Seems like you are only talking about in position on the turn.

When villain is calling less and checkraising more, you would barrel with hands that block his x/r range and have less equity (you don't care about having less equity vs call since most of your ev comes from fold equity, and villain won't even call you often). For example you would rather barrel Ks9h than Kh8h on As7s5h4h (assuming villain has high x/r on that turn). However if villain is calling a lot, or his strategy doesn't force you to fold vs a x/r, you can barrel K8hh more (can still barrel Ks9h depending on fold equity).

Theoretically (with thought experiment or with nodelocking in solvers) if someone never checkraises you would barrel almost all of your high equity hands (also a hand like A9o on A7ss54hh) whenever you have range advantage.

Assuming 100bb stacks, turn check-raises are quite rare aren't they? I'd be more interested in whether the turn card makes it more or less likely that villain will call again. If the turn increases my fold equity because it helps my range more than villain's, I'd be more inclined to barrel with my bluffs. If the turn looks like a blank, villain isn't going to fold much that called on the flop, so I'll more often take a free card with my higher equity draws, and mostly bet the weaker ones (fewer outs, less SDV) that can fold easily on the rare occasions that villain check-raises.

Thing is with solvers it's still correct to have a high betting frequency in some spots even if villain is x-raising you, it's just the highest EV strategy and it's just about appropriately defending in "some spots".

That Ks9h was sort of a bad example, but in general it depends on their range as well as their tendencies/frequencies.

Picking up Equity is a beautiful thing in poker, the question is too general to answer and you'd prolly have to write an essay to explain this question lol.