The ability to fudge rolls is the reason why I use a dice box. I peer under it and if I want to show it to the players I can (especially if you are using a hardback as a base for dice rolling).
Do not fudge too often...
Using the D&D Basic Game to introduce the players bit by bit to the rules is a very good idea. If they are very eager to learn, but are not willing to spend the money on the hardbacks, they can also use the SRD, but it is more cumbersome to learn that way.
Here is a good alternative for spellcasters. Try the Complete Spell Cards from The Other Game Company (that way the player does not need to page through the book to find a spell description - I think they are having a sale right now and it costs less than 5-6$). Second: If a player wants to play an arcane spellcaster, try to get him to play a Sorcerer instead of a Wizard. A Wizard may be more flexible and perhaps more powerful in the long run, but for learning purposes a Sorcerer is way better.
If your players are encountering an NPC, think about his personality (make some small notes about this, so that you are consistent, when the group meets him again - for example: Terseon Skellerang - Captain of the City Watch, JJJ, 6'5, had ***sing contest with player X). If your group runs into a part of the town or village where nothing is written down, make it up on the fly. Some real gems can come into existance that way. We had a gripping hunt over the roofs of a city (all improvised) because the group wanted to get out of town after their city guide had been poisoned right at their table in a tavern, the adventure had the conception, that the group would naturally stay at the tavern trying to solve the murder to clear their name...
Bring your players to play out their characters: use direct speech, when addressing the players as an NPC, and try to get them into the conversation. When the players start to use "I" instead of "The Rock" when describing the character's actions, you know you have won. When the player's are having an in-character conversation between themselves, sit back and listen (and have a hidden smile).
Next step (but that one is way down the road) is to bring your players to think about a motivation why the character is at this place now at this time and risking his life battling some monsters instead of being married and having to work for his/her income. Use the stories they come up with, weave them together with the other stories or some background material in your adventures.
Have fun...