Yes you need a spares kit. generally stick a spare head collar in. Hoof pick, pen knife, bit of string, Lead rope. If you have lamps batteries or candle depending on what powers them. note book and pencil, not a biro. Pencils work in the damp. I have a small human first aid kit in mine, a few plasters and a space blanket and triangular bandage. Spare breeching strap, trace and rein, if you want to go OTT.
What you wear is up to you. You can go uber posh and wear tie and posh hat. But whatever you wear you must wear a hat. You can wear riding hats, or ones with matching silks and match your groom or have your groom in smart riding wear.
Generally in exercise most folk I see go for smart casual with hard hats. Matching sweat shirts that kind of thing.But you must both wear gloves and the driver ought to have an apron.
I'd plait if the pony was of that type for a show class, not the tail though, just mane. Just present him as clean as you can, same with the harness and cart. Polish all metal parts.
Generally you will all go into the ring and trot and walk on both reins. Use whip signals in the ring if you can and try and drive coachman, It might make the difference of a couple of placings. The judge will line everyone up in any order then ask everyone to perform an individual show. Sometimes the judge will tell you what they'd like to see, other times they will ask you to do a show of your own making. Every time the judge has asked us to do a show it has included a rein back and lengthened strides. Sometimes a 20m circle with reins in one hand! A driving rein back should move the axle of the vehicle 3 metres in a straight line. Practice at home, keep it as calm as you can.
Just watch what other folk do and enjoy. When ever you stop get the groom out to stand by the pony's head facing him. He does not have to hold him, just stand in front in case they are needed.