It makes me feel like I'm just an idiot. I don't think I am. But maybe I am. Who knows. All I know is that I feel bad for killing all of those little guys. They didn't even die in the pursuit of knowledge. They just died for my own hubris and naivety.
The most efficient way to do this would be to just go sideways really fast from the ground. Problem: there's a lot of air in the way. So the right thing to do (again, both in Kerbal and in real life) is to initially go just up, but as the air reduces, gradually switch to moving sideways instead.
Then once you're up there, you need to know the basics of manipulating orbits. Lesson number 1: you can increase the height of your orbit on the opposite side of the planet by increasing your speed in the direction you're currently going. So once you get your apoapsis (highest point) above 100km or so, you can turn off your engines, wait until you get there, then burn hard in the direction you're going and your periapsis (the lowest point of the orbit) will raise up like magic.
You'll be orbiting in no time.
During launch, the only real timing or skill involved was knowing when to start turning east (around 10km, imho), and knowing how to keep your speed below the thresholds that air resistance was eating your delta-V (there's charts around to help with that, but it's not hard).
If you're using solid rocket motors that can't be throttled or stopped, you have a bit less control.
It's straightforward though, the documentation is here:
Even if you don't break out spreadsheets or game-specific calculators or whatever, you develop an intuition over time for what looks right as far as stage sizes, payload-to-fuel ratios, that kind of thing. I've done it all by eye & guesswork and have had successful out-and-back missions as far as Minmus and solar orbit (but my not taking it seriously enough and running some numbers ahead of time is probably why my attempts to reach the Mars-alike planet have just been big wastes of time—I just don't have the patience to figure out the ideal launch window and run the time ahead to then, to work out a proper budget for my fuel, and all that—plus those missions are easier with orbital rendezvous and I hate those, I can do them but they're such a pain in the ass)
As for killing all those little guys, consider adding a final-stage parachute that decouples the crew cabin from the rest of the craft. It might take them a few hours to hit the ground, but you'll despotically murder less space goblins that way.
The main thing is "asparagus staging", which is where you set up fuel transfer pipes between stages on the same vertical level such that the outside engines run out first, and the inside engines last. Say you have a single stage, Add two fuel tanks to the side of the main tank with separators and hook up a fuel line feeding fuel from the outside tanks to the inside tanks. Add engines to the outside tanks. Repeat for more layers, transferring fuel from each layer to the previous. Doing this can get you a rocket powerful enough to almost go in a straight line to wherever. Whenever an engine runs out of fuel you can jettison that stage, and have the remaining fuel tanks all entirely full.
The second thing is to make sure your final stage is as small as you can get it. Use a tiny capsule, a tiny fuel tank, and a tiny engine. This makes the final stage more maneuverable but more importantly it exponentially decreases the size of the rest of the rocket. As a rough rule of thumb each next stage should be around 4x the size. Once your rocket starts getting kinda tall switch to building a massive set of asparagus stages. Use struts to make the whole thing stop wobbling.
As others said, you need to go sideways to get into orbit. This translates to going vertically for 5k meters,slightly sideways (e. g. 10°) until 10k, then like 25° until 15k, then keep going until 30k, as you feel like you can go more sideways without making your rocket explode, do it. Do it by trial and error if needed, without trying to minmax, better to go more vertically and then horizontal. Consider that you want to keep going up so that your apoapsis (highest point of the orbit) is above 70k, which is when there is no air at all. You can notice that the "space" music starts only when you hit 70k.
Once the apoapsis is out at above 70k (ideally 80 or more), shutdown engine and wait in map until spaceship is about at the apoapsis. When that happens, point sideways (same direction you were always doing) turn on engine and keep them running until the periapsis get to 70k or more. If it's less than 70k, you will fall back.
That's about it!
To continue, start playing with plotting in the orbital map, it will give you a good idea of what you can do. Keep in mind that the way you measure how far you can go is by checking the spaceship delta V.
Good luck!
Thankfully there are plenty of YouTube tutorials that kind of summarize the basics.
Remember you pretty much have to go up and sideways to get into orbit.
at least you didn't say "pardon my French"...