Strengthening plant stand butt joints

If you use pocket screws for all the major joints here you don't have to have any particular worries about strength, they are very very strong if everything is done right¹.

Since this will see exposure to some weather (and the water from watering the plants I guess) be sure to use exterior-rated screws in the pocket holes.

How do I prevent it from tipping over?

Sufficient weight on the bottom platform will make this bottom-heavy so I'd expect this to be pretty stable, especially if positioned alongside a wall normally when it's not indoors. Since it's tall a particularly high wind could still topple it of course, unless temporarily secured to the wall by a length of chain or strapping when outside.

The base joint where vertical 2x2 piece joins the horizontal 2x2 is weak in my view. I was thinking of adding a 2x4 cross member (see zoomed picture) that attaches to both 2x2 pieces with pocket screws.

I'd definitely go with 2x4s or even 2x6s there for strength and stability.

Additionally, I was thinking of using corner braces at this joint

I'm not sure what you mean by corner braces here, but by all means add something² to give increased rigidity here if you feel it's needed. Rarely a bad idea to go stronger than you think is necessary in woodworking, until you have more experience and can better judge what's strong enough in a given situation.

Follow-on query from the Comments:

Any tips on selecting wood type? I assume pine would be easiest to work with for first project?

Actually as odd as it sounds pine might not be the best wood type to start with, since to begin with you'll be buying "pine" from most common sources — it may be called pine but will be generic SPF (spruce/pine/fir) and you can be getting anything from those families depending on where you are exactly, and what's in stock at the moment.

Even the size of the pieces you buy can change the species you're getting, from the same source. Larger boards are often a better wood simply because they have to come from larger trees, so they be cut from a better species (e.g. southern yellow pine).

In addition, this type of wood is usually poorly or only partially dried, because of the original intended use for the stuff — rough carpentry.

So be aware that milder hardwoods are essentially easier to work with overall, and almost always will be seasoned better. Poplar is a decent choice for the US since it's very widely available, cheap, quite soft and easy to work. Soft sounds bad but it's uniformly that hardness, not hard/soft in layers like in softwoods which have hard latewood and soft (sometimes very soft) earlywood bands.

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¹ Independent testing has shown that pocket-screw joints can actually exceed the strength of traditional M&T joins. There are worries they won't maintain this strength in the long term, but for stuff that isn't intended to endure they're one of a number of very suitable alternatives to traditional joinery.

² Gussets or diagonal wood braces.