How can I politely remind ATC that I'm waiting for instructions or clearance?
There's lots of advice I can give you here, but the answer is really that the degree of communication tower wants/needs depends on your local tower (and to some degree the individual controllers).

Perhaps the most universal piece of advice I can give you is to think like a controller.  Listen to the frequency and get a picture of what's going on in the airspace before you call them, and when you get good at it you'll also get good at knowing what kind of a situation you're flying in to (Is the airspace empty, or are there 4 planes in the pattern for landings, 3 in the opposite downwind shooting touch-and-gos, and a jet arrival on an instrument approach that just got dumped on the tower inside 7 miles barreling in &amp; messing everything up?) 

Also remember that from a controller's perspective of priorities ("Safe, Orderly, Expedient") speed doesn't count for much when compared to the other two requirements of the job (and aircraft on the ground are inherently Safe and Orderly unless they do something stupid like go charging across the hold-short line). Unfortunately for us pilots delaying the plane on the ground (that isn't moving, and is thus unlikely to hit anything) is generally preferable to trying to maneuver and delay the ones in the air (that can't stop moving unless they hit something, which is generally considered to be a Bad Thing).

For a masters class in "thinking like a controller" call your local tower and ask to arrange a tour / shadow a controller for an afternoon. Spending the day in the cab is a great way to get an appreciation for how crazy things can get up there.
You can usually Google up the phone number, or call Flight Service or your local FSDO and ask. For most towers it's relatively easy to set up a tour (they'll ask for a driver's license and possibly FAA certificate number to do a background check and make sure you're not completely insane).



The rest of this post is just my personal view on getting along with controllers based on doing most of my training &amp; flying at a pretty busy towered field (According to the FAA's ATADS "Tower Operations" ranking KFRG was #42 in 2013). 



On The Ground

When you're ready to go call tower and tell them - Something like Tower, Cessna 12345 number one runway 32, ready to go. is usually plenty (if your airport is configured such that there may be some question as to where you are specify which taxiway/intersection you're at).  

After you call the tower and tell them you're ready my experience is you're not likely to be forgotten about. You may be sitting there for a while (my personal record is about a half hour at the hold-short line), but the tower can usually see you sitting there at the runway end waiting. They want to get rid of you, they just need a hole big enough that they can get you onto the runway and airborne without creating any conflicts.

If you are forgotten (it does happen sometimes - you'll see other aircraft being cleared for takeoff who called the tower to report ready after you, or the controller changes, or you've been listening and are pretty sure the airspace is clear but you're still waiting) a polite call to remind the tower you're ready to go isn't out of order. Just don't be That Guy calling every couple of minutes, and if the controller gets a little snippy just be the better person and shrug it off.



In The Air

In the air is a slightly different story.  Sometimes it can take me 3 or 4 calls to get tower to even acknowledge my existence on busy days (again, you don't want to be That Guy calling the tower every 10 seconds - you should have a picture of what's going on in the airspace, and if the controller is busy you may have to circle outside their airspace for a while).  

Once you've made contact a good controller will communicate with you and tell you what they need you to do and where they want you to report, where to turn base, or who they want you to follow.

If you feel like you've been forgotten a well-timed position report is usually the best way to remind the tower you exist (and as a bonus it helps everyone else with their mental picture of the airspace too).
For example on an extended downwind for runway 01 at Republic you'll pass Sunrise Highway (a pretty easily identified road, and if you miss it you'll spot the train tracks right after it), the inner shore of Long Island, and the outer shore of the barrier beaches. Reporting each of those points while waiting for tower to call your base generally works (if they genuinely forgot about you they'll usually have an "Oh S%$^!" moment when you call them over the outer shoreline and turn you in for a 7-mile final).



One exception to my position reports technique is landing clearance -- I generally do something to remind myself if I've been cleared to land (fold a chart, turn my kneeboard, etc.) and if I don't recall being cleared I'll call somewhere between 500 feet and the threshold with something line Tower, confirm N12345 landing clearance? (Where depends on the conditions of the day - sometimes they crank out departures pretty tight &amp; if someone's rolling waiting until they're airborne is usually a good idea).
Sometimes they cleared me and I missed it.
Sometimes they didn't clear me and they do.
Sometimes they have an "Oh S%$^!" moment (they forgot about me, have a conflict, and I go around).

All of those situations are generally preferable to landing on a runway someone else may have been told they can use.