I have used an old cutting mat, when they are totally worn down they still work great with a sleeve over them

You can use a piece of thick tote interfacing (real stiff) or I prefer a piece of plastic needlepoint canvas (often called granny grate). You could even use a piece of foamcore or cardboard. I like the plastic canvas because it is waterproof. I cover them with a piece of fabric and then they can be removed when the tote goes into the wash or if they want to use the tote as a soft bag.

I made my sleeve for the bottom removable so it could be washed. I can also replace the bottom, if it becomes damaged.

My favorite thing to use for the bottom of bags is the plastic corrugated cardboard. I got several free from a shop that makes them, they were misprints. I have also collected political signs as they take them down after elections. I cover them with fabric. I have used the plastic canvas also.

I buy the plastic poster board from Walmart, make a long sleeve that is at least three to four inches longer than the bottom of the tote. Put the poster board in and fold the end under the base. That way I can wash the tote (with the sleeve pinned to the tote so they wash together and the sleeve does not join all the missing socks in the “hosezone”), as needed, and the plastic is not damaged. I did sew a piece of plastic canvas into the base of a purse I made, but when it was washed, the plastic canvas had crumbled from the heat of my home dryer. I had to remake the purse.....Grrr!

I have a problem with my tote/purse bottoms not staying in the bottom but coming up into the bag. Any suggestions on how to fix this?

Snaps can always be added - not a big fan of Velcro.

Sometimes heavy interfacing + whatever batting I'm using for the bag and sometimes cardboard covered with the lining of the bag (stitched up like a pillow case).

I have my husband cut a piece of wood, about an 1/8th inch thick into the size I need. He rounds off the edges with one of his sanders so it doesn't wear a hole in my purses or tote bags.

I use the acrylic from Lowes as well. It is a perfect solution for me. I also make a sleeve of the lining fabric to slip it in, seems to stay in place better. And when that bag has lived out its days, I save that piece of acrylic for the next one.
I know that Lowes will cut it for you, but did you know that you can cut it yourself? Just mark your lines, score really well with a box cutter, and snap apart. I use blue painters tape to wrap the edges and corners so it doesn't cut through the sleeve or bag.