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Like that title states what do you all use? I have seen masonite board being suggested, but there seems to be different kinds of that. So what works best for basing terrain pieces?

It depends how long I've got spare to spend on it. I've used MDF in the past, but recently if it's just and afternoon, thicker plasticard cut to size and shape suits pretty well.
In a real rush, thick card. But not for anything with large amounts of PVA.
I slap felt under whatever I make, out of habit, so the underneath becomes hidden anyway.

I will be using pva glue and basing material on the base, but what would be the most sturdy? Or would it all be sturdy?

PVA can cause some woods, boards and cards to warp because it's a pretty wet glue. They do flatten out again but it's seldom completely flat after.
Whilst they'd all be sturdy, only non porous boards or plastic will remain 100% flat. However there are ways to seal your boards or card to prevent warping if you have time to spare. I'm sure someone here would have a good explanation of how to do that, I've personally never done it.

Warping isn't just an issue of moisture, although any moisture in wood/paper materials will cause differential swelling, leading to (ideally temporary) warping. PVA shrinks as it dries, meaning any surface it adheres to will tend to curl inward/upward, even if no water is absorbed. Best counter is solid clamping of the base as it dries or simply using a different adhesive (construction adhesives like Liquid Nails don't shrink, for example).

Personally, I use 1/8" MDF for most terrain bases. A 4'x8' sheet from Lowes or Home Depot lasts quite a while, unless you're building tables. Decent balance of strength to weight/thickness, easy enough to work, and pretty inexpensive. Generally, I'll use hot glue (no shrinkage, good hold) to attach large elements, then PVA to attach basing materials, clamping/weighting the base to a flat surface as it dries.

A lot of my smaller pieces are mounted on old CDs. Bicarbonate plastic is light, tough and very rigid.

I have used Masonite (called hardboard in the UK) for larger pieces. It does have a danger of warping.

So it seems like MDF is the better route?

I'd say so, once you factor in cost. Nice, veneered plywood or styrene would be stronger (assuming an equivalent thickness) and a bit cleaner/easier to work with, but could easily run you four times as much as the MDF. If you have to buy materials, MDF is the go-to for most people, it seems.