I've finished my game for the wizard so I can outline the strategy which was working for me. I don't know what parts you know exactly, so I will tell you everything I consider important.
First of all, dump your quarterstaff and try to use daggers as your main weapons. They can be not only wielded, but also thrown, and if you become skilled in using daggers, you will be able to throw several daggers at once. Ten daggers or so is a good number to start with. Also I'd find an armor which doesn't hinder spellcasting (like studded leather armor) and try to enchant the cloak. Obviously enough, you should try to learn as many spells as possible and get all the resistances you can.
I advise not to go into the Gnomish Mines when you find the entrance. You can reach the Minetown easily, but you won't be able to reach Mines End yet and get out safely. So instead of going to the Mines, go to the Oracle and then to Sokoban. Sokoban should train your strength because you will move boulders a lot, which is helpful, and also you will get some gold and a gift at the end, which is, as you probably know, either bag of holding or the amulet of reflection.
After Sokoban you should go to Minetown and try to reach Mines End. Minetown has an altar and if you're lucky, it will be your god's altar. If that's the case, you should offer corpses of recently killed monsters to your god until you get Magicbane. This is a magical dagger, which is extremely helpful in combat.
Mines End will give you a luckstone (and also some gems). You have to fight a lot of monsters there though, so stand on the stairs and use Elbereth to keep them off. If you have Magicbane, engraving done with it will be harder to erase.
If you have survived Mines End, you can return from the Mines to the main branch and safely descend until you come to the level with the quest portal. Don't worry, if you haven't reached level 14, the first quest level usually has Wraiths whose corpses raise your level when eaten. If you still need to gain levels, descend lower in the main dungeon. Usually I don't linger in the level after I have explored all the passages. You should aquire necessary level before you reach Medusa's island.
When your level is enough, you can finish your quest. Your nemesis won't be tough if you know how to hinder his healing. After you finish him, you get your quest artifact which replenishes your power, and later on you shouldn't have any problems.
That's the basics of the wizard strategy.
Best Answer
How easy any given class is depends on your playing style of course, but most people agree that valkyrie is the easiest class in nethack. They're good for fighting, they start with good equipment (+3 small shield means that once you've picked up some body armor, you'll have a very nice AC for a starting character) and food and they have a decent quest artifact. They also get speed at level 7.
Race doesn't matter as much, but dwarves are a good choice for valkyries, because they have good stats for fighting and most inhabitants of the gnomish mines will be peaceful towards you (so you can go shopping without endangering your life - as much).