Some thoughts on speeding/simplifying play:
Round Display: an old, non-functional wall clock works well. Remove the clear cover and simply turn the minute hand from number to number as the rounds progress. Everyone can see what round it is. You can turn the second hand to point to an upcoming event as a reminder (e.g., if the circle of protection will run out on round 5, point the second hand to 5), or for multiple events, Post-its stuck around the circumference of the clock provide handy labels. The hour hand can be removed or ignored.
Have a "talking stick" turn indicator - I use a little plastic replica of the famous "Thinker" statue. When you are done with your turn, hand it to the next person in the initiative order. If there's an out-of-game distraction (e.g., the pizza arrives), when you get back to the table you can see whose turn it was.
Take 10 on initiative - PCs always have an initiative of 10+ their init mod, but you roll as normal for the monsters so there's still randomness about "Do I get to act before the monsters do?". It's easier for the kids to remember "I go after Fred, then Charlie goes after me", and you can seat them around the table in order of initiative, so they take their actions in turn from left to right (from your view).
Sorcerors and Favored Souls often work better than wizards and clerics for newbie players because the player doesn't have to track "How many magic missiles did I memorize?", they only have to track how many spells of each level they have remaining. In any case, make a little "spellbook" for each caster (Xeroxing from the PH or cut-and-paste from the SRD) that contains ONLY the spells they know - much faster to look things up in than flipping through the PH, and it avoids fights over limited copies (or the need to buy many copies).
Alternatively, "spell cards" can work well for wizards/clerics, each card having the name of a memorized spell and the necessary text. The player gets one card for each memorized spell (i.e., if they've memorized three Magic Missiles, they get three Magic Missile cards), and hands it over to the DM when they cast that spell. The DM can lay out ongoing spells on the table, with an attached Post-it indicating what round it ends, as a reminder of what buffs, etc. are active.