SCA standards say that at a minimum the armor must be 16 gauge in thickness; if you're speaking of mild steel, a lot of the fighters go heavier at 14ga and 12ga for their helms at the least. Mild steel is just that, usually cold-rolled, untempered, unhardened. Blued steel is just steel that's been blued or blackened, it has the same hardness or lack thereof as the steel it was before the blueing process was begun. The different polishes (satin, mirror, etc) only deal with how shiny you want your armor to be. I have noted a marked departure from the super shiny harnesses of the late 90s and early 00s- for myself, I prefer the satin finish. Its a good look w/out being gaudy. Stainless tends to be somewhat softer than mild steel which makes it less palatable for weapons applications but perfectly suitable for armor.
Spring steel can be thinner because it is harder by nature, and as such, it'll cost more. Then there's spring stainless which just means it won't rust as much but it'll be more costly still. Then there's spring steel, tempered - as far as I know, it is the 'Cadillac' of steels - it can be very thin, very light and still very protective. Not to mention, pretty expensive. You can get a full mild suit for about $2000, give or take, depending on how much plate you want. For the same suit in tempered spring steel, multiply by 5 at the least.
Of course, living history reenactors would and should shy away from anything other than mild or tempered high carbon steel; all else (stainless, spring, titanium, mithril, vibranium, adamantium etc) is modern invention.
Coats of plates...you said transitional period so you're looking for something along the lines of the Visby CoP; plenty of places to get those made, I've dealt with three different armorers and the results were more or less satisfactory, but I would only put forth two of them - Winter Tree Crafts, LLC and Mad Matt's Armory; MMa is in Canada though so the shipping charges may be of concern. WTC is in Vermont but he's scaled his workload way back as he's currently building a new house and shop on recently purchased land...he's worked me into his schedule but only as a favor to me as we've had good dealings in the past. Since a CoP is made up of plates riveted/sewn onto a leather facing you don't have to go high hog with stainless or super expensive steel; mine is being made of 18 ga mild steel plates- since they overlap, it'll be plenty protective. A CoP will run you $200-500 depending on armorer, materials, specs, etc.
You should probably avoid skimping- it will only lead to you replacing that part with a more expensive and better part, if you have the willpower, save yourself the trouble and go for what you really want. In this particular path of interest, as with most things, you get what you pay for and if you're needing it for its primary purpose of protection, you'll not want to skimp on it at all.
I was just thinking...you might get away with being skimpy on the undergarments; braies, chausses, under tunic - things not easily seen when in full kit or harness. If you have or know anyone who has skills in sewing, you can save yourself a good deal of money by working with patterns and sewing your own garments with the added bonus that it'll be to your specifications, not something off the rack that may or may not fit.