Far as I can tell, chainmail, chainmail and more chainmail.
This fella has a good look down and looks fairly SCA legal (depending on how his mittens are padded. He needs elbows as well.)
I have seen something along the lines of hidden springsteel mittens somewhere out there, if you could dress a pair in chainmail, you'd be set. What you should keep in mind is that at the start of the 14th century, there really aren't an awful lot of plate of any sort, not in vambraces, not in greaves, and greaves are pretty much the first piece to be improved in knightly armours. Metal shows up later on in the 14th, but for 13th, you're stuck in maile for England (to my knowledge).
EDIT: Here's a search of England, Scotland and Wales from 1200 to 1299
http://effigiesandbrasses.com/search/?y ... &genders=M
1244 is the first sign of any sort of hard metal, it being, possibly, a kneecop, since we see them on later effigies, but could simply be an un-mailed area above the knee, maybe of thick fabric to connect the thigh and shin mail and allow for flexibility and fit.
After this, there really isn't a whole lot of things on effigies, there are kneecops and they grow in popularity, one gent from 1294 have some sort of Eilettes or rondels on his arms, giving you a reason to have elbow cops with rondels on them if you feel like it, this man also has kneecops, but only regular hosen and turnshoes.
I hope this helps. You could always use barrel plastic inside of the channels of a gambeson, or linnen-covered hardened leather to cover up your forearms, thighs and shins in more rigid material than padded linnen underneath the chainmail.
EDIT 2:
Your first focus should be on becoming legal.
Especially considering the period kit needing a lot of tailored, riveted mail.
As such, I would suggest:
Helmet nr 1
Gauntlets 1.5
Elbowcops, Kneecops, Gorget nr 2
Gambesson, could be a bit thinner on your forearm to accomodate for a later addition of hardened leather vambrace covered in the same linnen.
Padded cuisses to prevent your drumsticks from a sound drumming.
Kidneybelt, cup.
Ready to rumble
Now, to build on this, add hardened leather covered in the same linnen you used for your gambeson over your forearms (unless plastic splints inside of the channels) and for your shins, this will make you look like a walking pajama, but it will also keep you whole.
Pajamaboy and ready to move on.
Chainmail haubergon, prefferably tailored to your measurements (with gambeson on), try to cover your gauntlets in the same rings if possible, bonus points if it all sits together OR if you can slide your entire gauntlet into the palm-slit of your mail mittens. NOTE: Mittens are a luxury, plate is safe, plate is fine, don't be too worried about plate gauntlets or demis and a basket hilt.
Final step:
Get your legs armoured up in chainmail, covering from toe to upper or middle thigh.
And you should be done. Take note that this is how I would go about getting this kit together and might well be something you are not interested in following to a tee, and might even be critiqued by other people with a LOT more experience than I have. The short version is get your helmet, get your rigid protection and your padding, throw a surcoat over it and go to town. Upgrade to chainmail later.