Everything ThreePhaseEel said, plus...
You need a main breaker in this detached building
Sorry, what you actually need is a "main disconnect". Further, if you're in Massachusetts, the main disconnect needs to be on the outside of the building because they adopted NEC 2020 already.
For everyone else, a "main breaker" is usually the cheapest and most expedient way to get a disconnect switch for a subpanel. That particular subpanel looks like it's built with space reserved for a drop-in main breaker. Consult with a competent Siemens dealer, and they're also the people to ask about whether your cover can accept a main breaker; otherwise they can probably get you a replacement cover that can.
If that just doesn't work (or you don't like the price), you can mount a 2-pole breaker in the normal breaker spaces and backfeed it. That breaker will need a tiedown strap.
The best grounding rod known is an UFER ground
That is a piece of reinforcing rod that sticks out of the concrete, then runs into/through the concrete and is tied to the reinforcing rod grid. Concrete is a fantastic ground plane.
What usually happens, though, is the concrete people never think of providing that, or the novice sees it, doesn't get it, and angrily hacks it off.
It is not a detached building if there's a breezeway
So if there's a breezeway, you don't need the disconnect or the grounding rods.
You're gonna need a bigger panel
Not quite yet, but eventually. You really don't have much expansion space left in that panel. I gather it was the right thing for present needs, but you'll soon be out. That'll happen sooner if you use a back-fed breaker as your disconnect switch, because you'll only have 1 space left.
"Oh, but I can just go to double-stuff breakers at that point" Not with NEC 2020, once that comes into effect (0-3 years), almost every circuit must be AFCI, GFCI or both. AFCI must be breakers ($40) and AFCI+GFCI breakers are only $10 more, so that's the economical way to do that. Your existing circuits will be grandfathered, but I'm not sure that'll allow you to convert to double-stuff then, since that removes any possibility of an AFCI upgrade, and you're not allowed to make a grandfathered situation worse. I can't believe I'm saying this, but it might be prudent to convert to double-stuff now while it's legal. Since you have to replace four of your breakers anyway due to being Square D.
Your two 240V circuits at upper left don't have a neutral, so they don't need common trip, so a quadplex breaker (double 2-pole double-stuff) is legal for them. Quadplex breakers don't have common trip on the outside, and some don't have it on the inside either.
Hey, since you need more spaces anyway, how about getting another subpanel and extending off this one. It could be Square D :) J/K it's only $30 worth of breakers...