I just went down this road about 6 months ago. Did a ton of research. This is going to get info intensive, so hang on.
You're going to need to use a safety razor. Or straight razor, but that's a whole other learning curve.
Modern razors with the 3, 4, and 5 blades only increase irritation if you already suffer from it. Each subsequent blade pulls the hair further and further out of the skin until the last one chops it off, where it drops under the skin surface and that's why you feel so smooth in one pass on the razor. The problem is that the hairs start to grow under the surface of the skin and some get caught on the way out and start to grow sideways into the pores of your skin. That and bacteria causes the razor burn.
Safety razors are different. One blade that only cuts, no lifting. You have to do more passes to get smooth (I do straight down, toward the nose, then toward the ear, but the hair is leveled at the skin surface so there are no ingrown hairs.
The biggest learning curve is to learn to only use the weight of the razor to shave and take short strokes. You really have to break the habit of pressing the razor into your face like you do with modern cartridge razors and trying to do long strokes. That WON'T work with a safety razor and you will pay for it.
In many ways, it's a lot like paint correction, several passes to level the surface, work small areas, slow arm speed, etc.
Things you want:
1. Safety razor - I suggest a Merkur 34C HD or if you have big hands get the Merkur 38C HD. Great quality razor without being stupid pompous artsy priced. Get the shortest handle you feel comfortable holding, because it's NOT a cartridge razor, you don't want to press at all! Longer handles tend to make new users leverage the razor into their face and cause problems. I got the longer handled razor as my hands are big. Just for reference, my hand span can reach past an iPad in landscape orientation thumb to pinky and it's one inch shy of the entire screen from bottom of the palm to tip of the middle finger in Portrait format. The 38C feels just right. If you hand is smaller, go with the 34C
2. Sampler pack of razor blades. The manufacturers of safety blades all have way different ideas of the ideal sharpness and angle of the blade approach. There is no way to tell what is best for you without trying a bunch. I started using the Merkur 10 pack as I was told that's as close to middle of the road as anyone could recommend, so I figured it was a good starting point. I used all ten blades before moving to a new type as I really wanted to get the shaving technique down before I tried to evaluate blades against each other. I bought about 5 different types for a total of about 40 blades, I've currently used 25 and 2 different types in 6 months. I get 4-5 shaves out of a razor blade. Oh, I also find that I stay smoother for longer into the day than with cartridge or electric shaving. I can skip days on occasion.
Pre-shave oil. At first I thought it was an optional item, but after using it, I'll never shave without it again.
After shave balm/lotion. Very important after using a safety razor.
Styptic pencil. For the oops that makes you bleed. Basically it feels like dumping rock salt into a shark bite, but it stops the bleeding in the 1/10th of the second that you can stand holding against your cut.
Other stuff like mugs, shaving soap or cream to lather, a badger hair brush, shaving stand, etc. are nice to have, but not necessary. You can use regular shave cream you buy at the store if you like. Me, I'm all in.
I've tried electrics, they don't get close enough or last through the day. Then I tried all the big name newest kid on the block razors and found that the more blades they added, the worse the irritation got. That's when I researched and found that I really wasn't alone in my observations.
Hope that helps you dude.
