I wouldn't worry about changing the tyres just yet. I'd make sure I'd got the items needed to change a tube/fix a puncture by the roadside (tubes, mini pump, tyre levers), possibly something to put them, along with a fiver, a bank card and a phone in - i.e.. a small saddle pack, plus a bottle cage and bottle. Maybe change the saddle for a
Charge Spoon then ride the snot out of it. (within COVID guidelines, of course!) Saddles are a very personal thing, naturally, but that looks like the standard Spesh 'body geometry' jobber on there and they can be, erm, oddly discouraging to ride!
There's plenty of videos on bike fitting - I'm a big fan of the 'your heel should just touch the pedal at the bottom of its stroke without tilting your hips' method of setting the correct seat height. There's lots of spacers under the stem and the stem is flipped, so the already pretty upright position of the Tricross is rendered even more so - i.e. there's plenty of room to lower the front end if needs be.
Spend a few quid - it's a freebie, after all - on making you and the bike self-sufficient then when you've racked up some miles, decide what to upgrade first and next. And have fun doing so!
(If it were mine, I'd rotate the bars upwards so the first section on the curves of the bar tops is horizontal, flip the stem, remove some of the spacers, remove the bar tape, move the brifters further forwards such the start of the rubbers flows almost horizontally from the line of the tops and the levers are easily accessible from the 45-degree section of the drops, and replace the bar tape with something not red... for the princely sum of about a tenner for the tape, but all of that is part of the fit process so entirely personal)