I like the fact that you're at least thinking of safety kit, so well done. Most, foolishly, don't even bother, so I think you're onto a good path already! Even if I was riding a moped, I would still wear protective kit. Sliding along the road at 30mph hurts the same on a 125cc bike as it does on a 1000cc sportsbike! Crashing is crashing!!
You only need a helmet - legally.
You say you only have a 10 minute ride, which I agree is short. However, it only takes a few seconds of sliding against tarmac to lose a lot of skin.
Here's what I'd recommend;
1. Brand new helmet that meets the ECE regulations and fits well
2. Gloves - protection in the knuckles, palm and fingers. Ideally with some high-abrasion resistant material on key areas (palm, fingers, side of hand). Maybe get two pairs - a thinner summer pair and a thicker, waterproof 'everyday' pair.
3. Boots - preferably over-the-ankle height, but you can get boots that cut off just above the ankle but still have ankle protection. These are more comfortable to walk around all day in - and typically look more like casual footwear.
4. Jacket - ideally waterproof with a removable fleece liner (so you only need 1 jacket all year round) and it MUST have ce-approved armour at the shoulder and elbows, and preferably with some form of back protector as well.
5. Trousers - textile waterproof trousers. Again with ce-approved armour at the knees and ideally the hips.
The above is the sort of kit I have to ride in.
The only extra thing you could get is a pair of Kevlar-lined jeans. These are brilliant! Indeed I only bought a pair to wear to university as I had a similar issue with carrying kit around all day. I wore the Kevlar jeans and a short pair of boots and they were all day comfortable! My jeans are RST and I would highly recommend them. I paid £80 for my pair but they fit very well, have armour and are made from decent quality denim (unlike some of the cheaper alternatives!)
If you perhaps cannot afford the RRP, have a look second hand. The likes of eBay and Gumtree can throw up some great bargains. Because biking is mostly a 'luxury', plenty of people get rid of them from time-to-time or barely ride them. This means that their hardly-worn kit has to go as well, and this is where your bargains come from! Always try kit on before buying it, but as I say you can get some branded kit for great value second hand. One thing I must stress though - ALWAYS BUY YOUR HELMET BRAND NEW FROM A SHOP AND GET ONE THAT FITS YOU WELL.
Good luck and ride safe
--Mikey B--