Fricking Google posted before finished
No table saw makes dowels pretty much your best bet. not sure what other equipment you have.


Pickets

Is there some other way to join these pieces that's not too "advanced"?

Not necessarily better but different and maybe easier.

- Use one dowel at the top end of each picket

- Use two screws through pre-drilled holes from the under side of the bottom rail

- Assembly hint, lay the side structure on a table and use spacers between the pickets to keep them equal and square, clamp up then drive the screws.


Legs
but I'm wondering what other options there might be.

Dowel at each end wouldn't hurt but could just glue them, no dowel or biscuit needed. Should be very easy to align as needed and if clamped well modern glue is all you need for that long butt joint. If you had a table saw you could do a kerf with a spline but alas.