Ummm I don't sail but have done a bit in the past and got friends who do, so might be able to help as nobody else has replied.
What sort of dinghy is it and what sort of sailing do you plan to do, i.e. pottering or racing? Sea or lake? Nice weather only or going-for-it? Have you done it before or complete beginner - & if so how do you plan to learn?
You prob need a buoyancy aid not a lifejacket - lifejackets are the big type with support round your neck, buoyancy aids are shorter, less bulky & let you move around better. Not sure where best to buy but have a look on t'internet or if you are serious about it look into sailing shops near you, once you start breaking bits of boat regularly etc you will be glad to find a good one.
Most people I know would only sail in either a wetsuit or a drysuit. Unless you are very much a fair-weather potterer with a very stable boat, in my experience you WILL get wet sooner or later and that means getting very cold if you wear normal clothes.
In a wetsuit you will get wet if you capsize, but the suit traps the water so your body heats it up and keeps you warm (up to a point!). But a drysuit is meant to be waterproof and has seals at the neck and wrists to try and keep water out completely. Under a drysuit you wear (old) normal clothes I think - make sure to wear enough warm stuff underneath if it's cold, as the suit doesn't give you much insulation, just keeps the water and wind off.
Generally I think drysuits are warmer (depending what you wear under), but a bit bulky/can be uncomfy or sweaty if it's hot and you are jumping around lots. They can also take a bit of maintaining if used hard, mending seals etc. (Not v nice if they spring a leak!). Wetsuits are a bit less bulky but less warm if you get wet (especially repeatedly) and obviously you will feel the wet. You will probably also need sailing boots if you get into it, I definitely recommend proper sailing gloves (to protect your hands from the ropes as well as keeping them warm), and a waterproof or spray top if wearing a wetsuit.
Remember lots of high-factor suncream as you're at more risk of burning on the water!
A good tip is also to join a local sailing club, they will have rescue boat cover for their sailing days, you can do races if you want and get the social side of it too.