2) Dungeon Map Next!
To start, you're only creating one dungeon, and that shuld occupy the players for about 2 or 3 levels.
Draw a dungeon map. I recommend making it at least two levels deep with a surface feature of some kind, so that's what I'm going to show here. You will need *three* pieces of graph paper.
a) On page 1, you split it into two maps. So fold your graph paper in half to make a little crease line.
b) On the top half you should make a vertical profilewhich shows the layers. The reason for this is you may eventually be expanding the dungeon, so leave some room for that.
c) the bottom half should show a surface encounter area- like maybe some ruins. Yeah, lets' go with that. So you have some overgrown foresty areas, some crumbling walls, or whatever. And you place a couple of low level encounters. Also make two entrances to the dungeon. One should be obvious (a big cave entrance and it goes right to room 1 on level 1) and the other one hidden and it goes right to level 2 somewhere.
d) Use the other two pages to draw your dungeon map.
* Make sure you include lots of multiple pathways- avoid bottlenecks.
* Don't bunch up the rooms too much. Make the rooms big enough to battle in.
* Make it good enough for you, because you're the only one who will see this map!
* Include a few traps. Put em right on the map!
* Make sure you put at least one stairway that leads from level 1 down to level 2 somewhere.
* Make sure to include that secret exit that leads out of level 2 back to the surface.
* Do at least 10 rooms. Preferably more.
You can also do sub-levels (little side-tracky mini-levels) if you are so inclined. Most importantly...
* have fun with it!
e) Now stock the dungeon. Since this is the 'first' dungeon the players will ever encounter, you have to be careful about encounters. You've already placed three encounters on the surface. Drop 5 to 10 (EL-1 or 2, or 1-2HD if we're talking about earlier versions of D&D) encounters on the first level. Stick an EL 3 in there somewhere too, near the back or the stairs. About half (or just over half) the encounters should have treasure*. I'd say make at least one encounter per level an NPC who isn't interested in fighting the PCs.
You don't have to write out full stat blocks for everything. Just write what encounter is there, how many, and anything else interesting. When it comes time, you can run the encounter right out of the Monster book.
* Rule of thumb is an average of 300 gp x encounter level. So if you don't give an EL 1 encounter any treasure, you need to give the next EL 1 encounter double (600gp).
Which ever dungeon this is, should be the one closest to the town you plan on having the PCs start in.