Improvising:
No matter how tight or how broad your preparation, your players are going to surprise you. You know it's going to happen. So don't worry when it does, and if improvisation is not your forte and you need to ask for a quick snack break while you figure things out, feel free. I've often found my players get sort of perversly proud when they do something I didn't anticipate and force me to make things up on the fly. A good tool for helping yourself with that would be to take a piece of paper and write half a dozen different items in a bunch of categories (land features, people you'd meet in a city, combat encounters, reasons so-and-so isn't home right now, and won't be til next year, etc.), so that if things go pearshaped mid-game, and any of those categories are remotely relevant, you can pick one or roll a die and have at least some idea of where to go from there. (South of the town is...a bear cave! The hooded figure is...Old man Wilkins! The Wizards' guild is currently...all chickens...terrible tragedy, that...)
A good rule of thumb with regards to mechanics that you can't remember is that if you can't find it in the book after a minute or two, make up a rule that seem reasonable and play with that. If you find it later, you can amend it, but try and get your players to agree before hand that they'd rather play for two hours with the wrong rules than for twenty minutes with the right ones. Also, getting sticky notes or bookmarks and putting them in often-referred-to sections of the rulebook is a good idea. Lastly, make players responsible for knowing their abilities, and hold them to the same time rule. Thumbing open the book to the marked page to check how many dice you roll is fine, fumbling around for several minutes trying to remember which chapter it's in isn't, and should cause you to, after a fair warning, either forfeit your turn or use a different ability.

Honestly, the amount of time necessary to resolve combat and handle intricate abilities and rules interactions is one of the reasons I gave up on DnD, so just remember that there are other systems that are less complex that can still let you roleplay as wizards and paladins. Just a thought.


Roleplaying:
If you are finding that your players seem lost in all things and are always looking to you for guidance, pick something that you feel is the most important for them to improve upon, and cut them some slack with the other stuff. DnD is complicated, and the freedom that comes with an RPG can be overwhelming if you don't know what to do. The fact that they're there shows that they're interested, but they are probably overwhelmed with choice. If you want them to get better with the rules, lend them your books, ask them to go over the rules, and then run some practices with them so they can get better (dream sequences for combat, or short, simple quests that drawn on various mechanics). If you want to encourage them to go exploring this wide new world, focus on their options and give them a hand when they need it to figure out the mechanics. They'll never get invested if they feel like they aren't doing anything right.

I said above that I think character motivation is important in character creation, and I don't think it's something that should be skipped over. If someone has no ideas, brainstorm with them, make suggestions and encourage them to come up with their own until something sticks. Don't worry if you get mostly unimaginitive steriotype, people often fall back on archetypes they know well when they don't know what else to do, and it's a stepping stone. Let them explore your world, but feel free to give them direction if they need it. As both an actor and a writer, I can say that being creative is a lot easier when you have some restrictions in place, and your players may find it easier to be creative in the way that they sneak past the guards to rescue the princess than to simply be interesting without any goal.
I've often found a good way to inspire my players to be creative is to present them with an obstacle between them and their goal that I haven't bothered to think of a solution to. I'm just careful to not outline it in too much detail, and then I allow my players to create the solutions themselves ("How high is the wall, can I jump it? Is there anything nearby to climb? Are there small cracks in the wall I can use to scale it? Could I jump across from the nearby building?" "Uh...yes, you do notice small cracks now that you look closely, and it does occur to you that the nearby building has stair you could climb."). Of course, if you just made up a problem for them, you need to let one of their solutions work, or be able to come up with some other way for them to succeed.

I don't usually talk in character for all my NPCs, as I find it gets wearing and I stumble over what I'm trying to say, but I will jump in from time to time and address players directly, in character, to try to draw them in when they're describing their actions in broad terms. ("I ask the barkeep if he knows about dragons." "Oh, Dreggens, yeh say? Now whit would a good-likken fellah lik yersel be wenten with Dreggens, I wonder? Eh? Whit do yeh say, boy?") Expecting most players to keep up in character dialogue all the time is, in my experience, asking too much, but drawing them into it at good moments can enhance their involvement. Likewise, if they say "I pick the lock. Does it work?", you can either take the time to do a detailed description of the woodwork and the lock and the feel of the tumblers, with a prompt for them to narrate more, or you can just ask them to describe in more detail what they are doing, which is especially relevant in situations where other people might be watching them. Again, doing this too much will wear your players out, but demonstrating some good descriptions and them prompting them to provide their own ought to drawn them in more.
Lastly, some players are taciturn and neither well suited nor inclined to giving detailed or creative descriptions, even if they are otherwise having a ball, and you may have to resign yourself to having a player or two who simply won't engage quite as much. As long as you are both able to have fun.

There is also the mechanic of plot points, which you can steal from the other systems that use them. You can implement them how you like, but they are basically points that are awarded by the GM for good roleplaying, clever or inventive thinking or other actions deemed meritous, that can be cashed in to reroll dice or to subtly tweak the story ("No, I totally remembered to bring that seemingly useless scrap of paper." "Nope, definitely prepared Explosive Runes today. Yup." "Why, I recognize that fellow, he's my brother-in-law!"), of course such tweaks are subject to the GM's discretion and can cost many points, depending on the tweak. I also usually represent them with actualy sweeties, and thus allow my players to eat their earned plot points.


On the subject of maps, I find they usually take some time and all of the group's attention to draw, so I only bother with them for locations that are both important and potentially ambiguous. A narrow tavern or the throne room ought to be easy enough for people to envision, so don't waste the time, just go with a good description. A chase through a crowded warehouse with a maze of heavy boxes, however, may require a diagram so as to avoid confusion. Likewise, miniatures are cool if you have them, but hardly necessary, and I usually just sketch the enemies onto a map if I'm drawing one, and again only if I feel a map is needed. Usually, most combat is linear enough that it is straightforward, or sprawled enough that it is hard to draw anyway.


Hope some of that was helpful advice, and that you were able to make it through it all. Let me know if there's anything else I can say, or if any of this ends up being of any use to you.