I leveled plenty of random stuff, but most of it was combat related.
Skyrim is so much more forgiving about leveling random stuff than Oblivion is. It’s still possible to screw yourself though. If you level up a huge amount of skills inapplicable to combat like pickpocket, lockpick, and speech, then you’ll face stronger enemies but won’t be stronger at fighting. Another mistake is to neglect the health stat, that’s the primary stat you want.
I wanted to be a magical stealth archer at first then I found stealth too slow to use all the time and in broad daylight outside people saw me from far away a lot. Then I wanted to master a wide variety of magic to try out a whole bunch of spells.
So it was only eventually that I really got into heavy armor, one handed, and shields. I found that they served me very well. With a shield I could charge people over, block attacks, resist magic, and stun lock opponents. Block was OP. Then pair that with the ability to quick switch a shield away for a healing spell and I was nigh impossible to take down.
Obviously most damage comes from one handed. You’ll want the damage increasing perks from that. You’ll want damage effects on your weapon (like absorb health) that never run out (from having 100% cost reduction on that magic type on your armor). You’ll want a weapon that has been super improved at a grindstone while wearing fortify smithing gear. You’ll want fortify one handed on your gear.