Should I buy that spanner?
Check whether your current pedals have got an allen key fitting in the axle end pointing towards the frame. If not the person changing the pedals will need a spanner. Given you can't even replace a gear cable yourself, I'd get the LBS to do all this (and don't buy the spanner).
For this bike there's only one choice of tyre to do it justice:
Continental GP5000s. They've got really good graphics on the side, they'll fit the wheels/rims really nice and tightly, and their rolling resistance is right up there with the best. By no means cheap (but you'd get 3 tyres for the price of the shoes). Probably
25mm wide to fit into the frame. Changing the tyres over is certainly something you could do yourself, and take pride in doing so (tyre levers and pump required).