Looks like botryoidal agatized Florida coral to me. BTW, welcome to the board.

Yes, two nice big pieces of Pet Wood ready to be finished for bookends in the back.
The piece with all the swirling circle patterns in front of the Pet Wood looks like the famous and rare Luna Agate from Mexico, although it would be nice to see it wet to see the colors better.
And as RickB said, the bumpy brown stuff is called Botryoidal Chalcedony, a fairly common agate formation when the rock has an opening to fill. But those are larger 'bumps' than you usually see. We would have to see more of that whole piece to say for sure but I really like RickB's guess about it being petrified or fossilized or agatized Florida coral.
If you would like us to guess on the 'green' piece, left side of group, we will need better pictures from a couple different angles. It might be a local rock to you out there?

Hey Graham, To post more pictures,  to the left of the Browse box, just hit the little arrow after the word "Attachments" and another Browse box will appear. You can do that multiple times per post.

And I do believe RickB nailed it on the coral. It is called Withlacoochee River coral and starts in Southern Georgia and goes thru Florida.
Here's a link with pictures to compare;
https://www.thefossilforum.com/topic/110534-agatized-coral-from-the-withlacooche...

Very hard to say which way would be more "valuable", Graham.
It's more rare for that coral to be solid inside. Most of those pieces do have a hollow cavity and there could be some interesting formations. But will that make it more valuable? It's a real nice specimen, with very large pretty round bumps on it. Might be able to get $50-60 for it as is.
Then again, if you cut it in half, and it's hollow you might get $80 or more?

It's a very pretty specimen as is and I don't know if having two halves would look any better? On this piece I would say the main reason to cut it would be to see and satisfy curiosity. But then you have to decide which way would be the best way to cut it? Might want to set it on the shelf for a while, take some offers, and cut some other rocks. You're getting a good lesson about being between a rock and a hard place!

Glad to hear it, Graham! By all means, if you plan to keep it for yourself you will eventually have to know what's inside! Good luck and please show us what you find.
Thanks for the picture of your beautiful unique creations. Well done!

Well I just have to figure out a way to clamp it in the saw !