Yes, you get the travel reimbursed, it comes through on a separate payslip. Your DIM (manager) will claim it for you on the system.
Do NOT wear wellies. Wear your most comfortable grippy trainers until footwear is issued. Your legs will be aching for the first couple of weeks. As for cutting off, for the rest of this week you should be out with a pair of posties, so in fact you'll all likely finish early with the extra pair of legs.
You don't work a week-in-hand so you'll be paid this week. If you last 6months your pay automatically goes up 50p an hour, then again after 12 months. Tights will be ripped to shreds after manoeuvring yourself between hedges and parked cars on drives, scraping down walls through 18-inch gaps etc. plus the obligatory overgrown garden with trailing brambles.
You'll likely be kept on the same round for first few weeks until you get up to speed. Then the stress starts as you move to new frames with new partners each week, or even a few times a week.
Be prepared for some posties to be impatient with you if they think they won't finish early because you are new and slower as a result. Most are friendly and understanding but when push comes to shove it's each man for themselves.
Join the union after say a month or 6 weeks when you know for sure you'll be able to hack it.
And good luck!
From Posties posting Dog Turds through some awkward customers letterbox to customers asking you if you can cut their toenails for them! 
One of my favorite Guns and Roses tracks 