Reddit mentions of Clubman Woltra Styptic Pencil Small Twin Pk. (Blister), 0.25 oz

Pro tip: get a styptic pencil (aka shaving pencil). Makes it clot pretty much instantly.

Check out a Merkur safety razor (https://www.amazon.com/Merkur-Long-Handled-Safety-Razor/dp/B000NL0T1G) and a sample blade pack from amazon.

Get an inexpensive shaving brush to start with to see how you like it, but eventually drop more money on a nice brush. Spend $10-$15 to start then expect to drop $50+ on a nice pure badger hair brush in a few months. The badger brush that I bought for like $70 I still have with me 4-5 years later (although admittedly I don't shave terribly often). Once you get that pricey brush, get a cheap plastic holder for it as well - keep the bristles pointing down instead of up - after using it leaving them facing upwards allows the water to settle around the base and weaken the glue holding in the bristles, ultimately destroying your expensive brush. Not good.

Check out Proraso soaps in a bowl - more convenient than paste or anything else, I've found. They have multiple types, here's my favorite as it makes my face a little tingly: https://www.amazon.com/Proraso-Shaving-Soap-Refreshing-Toning/dp/B00837YY18

If you prefer your own bowl, Proraso makes a paste and you can get a mug or bowl to mix it in - I found I preferred the ready-to-go stuff as it's faster and makes it more likely I'll shave regularly. :)

Then, after you've done all that and realized it's the best shave you've ever done and it's actually way more fun to shave than you ever thought it'd be, go for a straight razor. Don't do an actual blade, strop, and all that stuff right away. Instead, just go get a disposable straight razor blade holder and some blades: https://www.amazon.com/Equinox-Professional-Straight-Single-Blades/dp/B0118BJ0PA and learn how to use that thing. Then if you're like me and shave once every week or two (super lazy!) you can literally take weeks of beard off in a single pass with a straight razor and a fresh blade. I stopped here, personally, didn't go on to a full straight razor as they're pricey and you need to maintain it (oil, strop, etc) - likely something I'd let slide and ultimately wish I hadn't spent the money on...

Also, get a styptic pencil: https://www.amazon.com/Woltra-Styptic-Pencil-Small-0-25/dp/B000EGIEOE - it stings a bit when you cut yourself but almost immediately stops the bleeding. Unlike cuts/nicks with a multiblade razor, cuts with a safety razor or straight razor are actual "cuts" - and they bleed like cuts. Like, "blood trickling down your face" type cuts. They look worse than they are because you have water on your face and it thins the blood and it runs more, but it bleeds. This will stop that bleeding really fast at the expense of a little more pain right up front. I keep one handy.

I've turned several friends on to at least the safety razor. It's way more fun to shave that way, less expensive, and is better for your skin to boot. If you like a really close shave, too, you can get a better shave overall with a 3-pass technique (I don't do that, personally) as you get better. All sorts of options...

edit: mentioned the brush holder
edit2: styptic pencil

Could it be a styptic pencil? Usually found in the bathroom or 1st aid kit -- you moisten it with water, and then use it to stop bleeding from minor nicks and cuts, like when shaving for example.

Example

I got started with safety razor relatively cheaply, buying everything on amazon. I'm linking to individual items, but you can use amazon's suggestions to bundle most of this together for less than $35. And the best part is, soap and blades can cost less than $20 a year.

My razor + first blades

You'll need this (especially as a beginner)

These can probably last you a year or more.

You can buy regular shaving cream at first, but your curiosity will lead you here.

I am making the switch too. After reading a lot on here and talking with a buddy who uses a DE this is what I ordered (should be arriving today). Total cost was around $125. If anyone knows anything I should add let me know. (Edit formatting)

Double Edge Safety Razor Blade Variety Pack

Merkur Model 180 Long Handled Safety Razor

Taylor of Old Bond Street St. James Shaving Cream

Nick Relief Styptic Pencil

Colonel Conk Evernice Model 775 Chrome Safety Razor Stand

Tweezerman Men's Shaving Brush

Proraso Pre-shave Cream

A Styptic Pencil for when you nick yourself shaving.