Most of the hard dungeons are really not all that bad, but a big step up from the easy and normal ones for sure. Knowing how to play DD in general really helps with the specific challenges as well. The wiki has some good strategies
for the alpha, and most of them still apply and are quite helpful.
Some general tips on game flow:
Usually you first want to see what's what: check what altars are in the game and start worshiping a god, find one or two level 2 enemies you can beat to get to level 2, get level 3, find some helpful glyphs and powerups, etc. Usually you want to use altar scouting and rush straight for the altars, then worship the first useful god you find. If you want to get around some annoying conducts it is usually better to worship asap and switch later than to delay worship. If you are worshiping MA you want to start building piety for mana leak asap, and don't waste any mana regen.
In phase 2 you want to get as much bonus exp from killing higher level enemies as you can, and pick up some helpful boons from your god. Storing up piety for heals or protections against the boss and the like is another goal in this phase. Killing enemies one, two or even 3 (rarely 4 is possible) levels above you gets you increasingly nice amounts of bonus exp. Getting a high level helps with the boss fight, and getting it with as few kills as possible also helps. Stray low level enemies that can be picked off without retaliation for bonus exp and piety are extremely helpful, so leave them alive if you can. You also need to find the boss now if you haven't already, and in most levels you may explore most of the level, get all the potions and powerups, and find all the glyphs (and convert the ones you are sure you are not going to need). Having a few tiles left to regen up before starting the boss fight is highly advised. Also you usually don't want to use potions in the first two phases.
At level 7 or 8 you want to start fighting the boss. The usual strategy is to get almost to the next level, then deplete your mana and HP (but not spend any potions and piety) fighting the boss. Then you heal to full by killing some level 1 or 2 exp bag that cant even fight back. You are committed now, no turning back. Before spending the levelup you can abort and try again later, afterwards probably not. Now the fight proper begins, try to optimize your glyph, piety and potion use as much as possible. Optimizing glyph use includes converting glyphs you don't need anymore. After the boss fight you are done, so don't save anything for later. Switching gods at some point before or during the boss fight might be useful, especially if your current god doesn't like you using potions or certain helpful glyphs. Desecration can also be very helpful, especially of the MA or TT altars. Desecrating GG, Taurog or Dracul altars is only advised if you know exactly what you're doing.
Good preparations are tri-sword (don't convert this, it's hard to get back), altar scouting, the magnets (which one you want depends on the level, fireball is almost always useful, and PISORF and ENDISWAL can help with enemies blocking your exploration), smuggler's den and the 25 gold from the bank. Also you usually want to bring the sword or shield from the smith. Shop scouting is nice not so much for the shops but for the general scouting. It may reveal glyph or enemy positions and walls which can be very useful. Altar scouting reveals a slightly larger area than shop scouting, use that to tell them apart.
A really good class that hasn't been mentioned yet is Gnome Warlord. This one got hit hard by the god nerfs though. Taurog, TT and Dracul don't really work well with this one, and MA and GG got nerfed. Still good though, especially if you get MA with BLUDTUPOWA.
For some more specific advice here are some good classes to beat certain dungeons with (the ones I like). Of course others work too, and these may very well not be the best ones. Also some dungeon specific strategy tips:
Havendale Bridge: Orc/Halfling Berserker. Don't pay the troll, kill it. You may have to use a mid-fight levelup to do so, and you're going to need some glyphs. Maybe even a potion or two and a boon from your god. But conversion is extremely useful for an orc, so you want that sensory stone. If you get PISORF or WEYTWUT you can get the troll out of the way and kill it later. Useful gods are mostly just Taurog, Dracul, and possibly TT. The other two really take more than they give here. MA can be marginally useful but don't expect great things from her. The boss is not particularly difficult thanks to your maxed out magic resistance (with 3 charms). Kill high level druids to level up. You may have to restart depending on what stuff you get on your side of the bridge.
Druid Grove: Orc/Halfling/Human Berserker. Kill druids to level up. Imps and Bandits are also fairly easy, but Druids are far nicer because of their magic damage. Taurog is by far the most useful god here. The boss is easy to outregen with high magic resistance, but the plants he spawns tend to get in the way of exploration. It's better to kill him in as few hits as possible. Note that dangerous plants may block the path to some enemies, your altar, and even the boss himself, so don't take access to these for granted. Don't be afraid to kill resist lowering plants if you have to, and try to keep death protection on you.
Creeplight Ruins: Halfling Priest. Leave the cultists alone until after you turn them into shades. Most gods suck in the current version, and Dracul doesn't like healing potions and killing undead, so once again you are best off with Taurog. MA can also be good if you find CYDSTEPP or HALPMEH. TT is also fine. The god you get affects your strategy pretty strongly, so adapt accordingly. Generally you want to explore a lot of the level before slaughtering the sacrificial goat to avoid lifesteal from the shades, and so you don't get screwed so much by their blinking. Kill zombies in the meantime. The boss has 3 forms. The first one does a lot of magic damage, you want some magic resist and lots of HP to survive, death protection is very useful. Also a good fireball target. The second form is undead and magic resistant, the easiest one for a priest. The third form is basically the regular wraith boss, with mana burn and 60% physical resistance. Physical attacks work fine against this one for a priest, but it also makes a great fireball target if you didn't take too many of Taurog's boons yet or are using a different god. If your level is high enough you can just alternate potions and physical attacks against all three forms. This will easily be enough to take the boss down if you have a tri-sword. If you didn't bring one some god-related shenanigans and mid-fight leveling are needed.