Should I get it to a real trellis soon then?

A trellis, cage, or stake would certainly help it to find the support it needs.

If you are new to tomatoes... although tomatoes are technically vines, they won't put out tendrils to 'grab' the support structure like other vines. You'll need to make sure they are tied to stakes or trellises. Make sure it's secure, but not so tight that the stem gets 'choked' or girdled.

I would bury the tomato deeper in a larger pot, roots will develop from those nodes and you will get a healthier plant. However, you need to watch the watering or the stem of the plant could rot.

The leaves had been drying out, so I upped the water. This time I just did way too much, but it is not typical.

Turtle, unless I am mistaken, Max means that if you transplant into a garden, or a larger pot, and you bury the stem a bit to encourage those root nodes to find soil, then you may have to be careful not to over-water or perhaps the stem may rot before the roots form.

I've personally never had that problem, but I suppose, especially if drainage is not optimum, it could happen. Tomatoes are notoriously prone to fungi, so, care is certainly appropriate.

Most people who keep tomatoes in containers find they do much better in as large a container as possible.

I will be moving it to the planter that is in the rest of the questions I have put on the garden forum.