Awesome game. Funny and entertaining. I've beat it over a dozen times now and still come back to pass time.
For anyone having difficulty with Nethack (or other difficulties in general, took me quite some time to beat it on Easy my first time playing), I find it the easiest to beat with Paladin. Just keep donating to the Church of Quilogy. Dex is your heal stat and Armor doesn't matter for melee classes (even if it goes into the negatives), they get protection from their equip. Power increases enough on it's own each level-up so that you don't have to bother donating to the Church of Kilogy.
Each use of Heal only costs 15gp, and heals (Dex*5), compared to the potions costing 20gp and healing for (level+2)*10. As long as you donate to the church of Quilogy three times (25gp each), your Dex will increase enough each level-up (+2) for the Heal skill to out-heal the potion, and for less cost. You don't rely on shops to stock up, so you can spend your money freely in upgrades since you won't have to worry about saving money for the next potion shop. Just make sure you always have like 45-60gp on hand if you meet a tough enemy. Remember = no money, no Heals during battle.
Just keep buying the upgrades as they appear (I prioritize Equip, so I need to spend less money casting Heal, but once you get the blue armor that decreases damage by 10 you might want to start focusing on the swords so grinding goes by faster), and grinding in the Dungeon. I've reached level 25 there, but it's pretty slow to build exp past level 10 or so. After you reach level 8-ish, you should be strong enough to one-hit enemies if you have the best weapon. Just keep grinding. I don't like leaving the Dungeon until I'm at least level 15, and have 50 in Dex and all the weapon/equip upgrades bought. You'll be hitting the 999gp cap pretty often as you grind, so I find Paladin easier and safer to farm money with than the Rogue.
For the second area, you'll probably be one-hitting people too if you grinded enough in the Dungeon. Just make sure to bring your Dex to 100 before moving onto the third area. I remember missing Randal when I had like 70, and you don't really want that to happen. You should be one- or two-hitting everybody in the third area too if you're high-level with all the best gear, and 50+ Power (you get 40 from leveling alone by the time you reach level 20).
Now, if you're doing a fresh game and having trouble getting any class to level 10 to unlock Paladin, I also have some tips:
-Play on Easy, you can do Normal/Nethack later.
-Do it in the Dungeon, there's no need to move to the second area to get to level 10.
-Stock-up on potions, none of the starting classes have self-healing, and playing on easy won't help if you need heals mid-battle. Only buy upgrades if you have 5-ish potions stocked up (10 is a safer margin, but lack of gear might be deadly on it's own).
-If you're playing Warrior, just spam Bash and forget the other skills. Again, donate to the Church of Quilogy to raise Dex and lower Armor, and feel free to donate to the church of Kilogy to raise Power and lower Dex (but not *too* much, you'll still want like 20-30 to hit people in the Dungeon).
-If you're playing Ranger (I was the first time I managed to beat the game), just spam Twice on the weaker enemies. Rangers actually get more HP than Warriors (25 vs 20 per level up), so focusing on Armor by lowering HP (Church of Defolgy) then spamming Once is actually a valid strategy. You can even sacrifice Dex for Power if you don't mind sitting around trading arrows for 10 rounds until you actually hit (people won't hit you if you have 20+ Armor and keep using Once).
-Esper is easy to take to level 10, but you'll need a bit of luck to run into upgrades in the early game (I prioritize Equip first). Sense and Focus don't take your turn, so there's no real reason not to spam Sense before every attack until you run out of PP. Only use Focus if you've fought the enemy before and know you can't hit it with your current Dex. I like focusing on Power with this class, but Dex shouldn't be sacrificed too much, since max PP and Focus works off Dex, so it might not buff you too much if you can't hit normally, and you'll run out of PP fast overall.
-Rogue can be used to cheese some fights (mob with gas can, for example) and get lots of money to be used on upgrades and potions, but I find it less reliable than the alternatives. If you want to go for it, just use Steal against weak enemies (it deals some damage), and Counter+Stab against stronger ones. Focus on buying weapons, Cloak only seems to affect Counter, not Steal.
-I'm not a fan of Wizard or Monk to unlock Paladin with. I'm pretty sure it can be done easily enough, but I don't have a strategy written down, and I don't like how they need to waste turns to recharge/use their items.
-Cleric is...viable. Not the best, since it requires luck and some minor effort to unlock, but if you also hate not being able to self-heal without potions you might be tempted to try it. It's far easier than Paladin to unlock, just have 10 potions on you at once (Rogue is good for this one). In combat, you'll pretty much have to Heal>Smite>Recharge every battle. Focus on buying 1 upgrade for the staff and 2 for the gem as soon as possible, and change the strategy to Smite>Smite>Smite>Heal>Recharge (one less smite if you only have 1 gem upgrade) vs weak enemies, and Smite>Heal>Heal>Recharge against stronger ones (not possible unless you have level 1 staff and level 3 gem). The luck comes in getting these upgrades before you start encountering Dungeon Wanderers regularly. They'll kill you pretty quickly if you can't heal twice per turn and still attack. Always focus on getting the staff upgraded once, then the gem upgraded twice after, so you can get one more Heal (or two Smites, vs weak enemies) per Recharge. Dex is also your Heal stat, but each point only adds +1, not +5 like Paladins. The upside is that each staff upgrade adds +10 or +15 (last upgrades adds +25).
-If you play the Flute three times in the same playthrough (high Dex is needed to succeed, event only encountered in Dungeon), you'll unlock Bard. This is a fairly easy class to get to level 10 as well. Just use Legend once then spam Mock every fight. Power and dagger upgrades are useless if you keep using Legend, so invest everything in Dex and only buy Flute upgrades.