Hey that's very cool you're doing silver! Firstly I don't know what organisation you're doing DofE through (ie your school or an outside group etc) but often they'll provide equipment like tents, basic cooking equipment and a first aid kit, so it's worth making sure you know what will be provided. Also worth communicating with your group - there are some things eg washing up equipment where you only need one for the group, so make sure you decide who brings what.
Things that you should bring yourself are a rucksack (there's a lot that can be said about having the right bag and I'm not the person to say it), sleeping bag and sleeping mat, clothes (hiking clothes are good, and I'd recommend having a fleece and a waterproof jacket. My school also insisted we brought waterproof trouser covers) - don't worry about bringing too many clothes, it's not a very long trip and you want to keep your bag light where possible. Hiking boots (make sure they fit well, you'll be doing a lot of walking). It's also worth having a small first aid kit (or even just blister plasters. Definitely bring blister plasters) and a torch. Finally, bring the basics in personal hygiene (toothbrush, toothpaste (although if you really want to travel light you can probably bring one for the whole group) and deodorant).
You'll want food as well - coordinate with your group on this, as it's best if you can split food up among the group to make best use of space/keep weight down.
I'm afraid I can't help much with where the best places to get equipment are/the benefits of renting vs buying. I suppose it depends a bit if you think you're likely to use things again. If you can borrow a sleeping bag/mat or even a rucksack then that can be worth doing, but having decent boots is definitely important. When you started DofE you should have got a welcome pack (it'll be in a silver envelope and contain a little booklet of the assessor report forms). There should also be a little plastic card that entitles you to at least 10% off at various shops (Go Outdoors, Blacks, Millets, Ultimate Outdoors, Tiso) so it might be worth going to one of them as you can get things at a discount, and DofE has clearly deemed them reasonable quality.
I hope this helps and good luck on your expedition!!!