@BigMeatball and
@LemonJuice, come on now let’s not fall out and loose a member on the forum.
@LemonJuice, over the 14 pages so far, your bike will be serviced and put back on the road by the mechanic but you will supply a new 8 speed cassette and 8 speed chain and are looking at getting a spare 8 speed rear mech.
If you are keeping your shifters then you need a Shimano R2300/Claris short cage (SS) or medium cage (GS) but not a long cage (LS). If you get the R2400 version it is just the new version of Claris which is the same as R2300 but the shifters have hidden cabling unlike the R2300 which has the gear cabling coming out the side. If you do upgrade to the shifters you will need a new front derailleur as the ratio I believe is different and 2300 and 2400 are not compatible.
As others have said you do not need a new rear mech as the spring in it currently has the incorrect tension and when the bike mechanic fixes it it will look fine and correct. He will most likely loosen the cable and retighten under tension to fix or if the cable is fraying he will replace it.
Cheap items like cables, cable outers and spokes are not worth supplying yourself as the shop will bulk buy and keep as stock so they most likely can supply for cheaper.
For shoes and cleats and pedals buy them secondhand first to see if you like this way of cycling and are comfortable with it; use it as a chance to try out SPD, SPD-SL, Look Keo, crank brother egg beaters or Speedplay first and then sell on and invest in a proper pair of pedals and shoes which you know you will like. For me I can’t get on with SPD-SL so use SPD or look Keo on my bikes.
If you do supply parts yourself the shop will normally charge labour time plus £5-£10 but if they supply the charge is usually waived and in total in can work out cheaper than the individual sum of the parts.