I'm a student pilot who has just begun dual cross-country with my instructor. However, he has not been explicit when I've asked him about radio comm with tower/approach/radio during the flight. If I enter an airspace (say, Class D) and contact the tower, but I do not intend to land there, what would my initial call to the tower be? And what are the differences in requests for approach/departure and tower?
It's the standard initial contact: who you are talking to, who you are, where you are, what you want. "Madison Approach, Cessna 123AB, 15 miles south at 3000 feet, request to transition your airspace to the north." If it's class B, C, or a TRSA your call will be to Approach and they will give you a squawk code and the local altimeter. For class D, there is no approach so call the tower with the same request.
This will vary for every airspace so lets go over them in descending order.
Class B (Bravo): First off, keep in mind for your future solos that as a student pilot you are not allowed to fly solo in class B airspace unless your instructor has given you training and a logbook endorsement for that specific class B (see 14 CFR 61.95). But if your instructor is in the plane then technically they are accepting the clearance, not you. For Bravo airspace a clearance is explicitly required and a typical exchange will go something like this:
PIC: "1234XY 5 miles south west of KPHL at 4000 heading to KPNE requesting transition through the airspace."
ATC: "1234XY squak 3425 and ident, altimeter 29.92"
PIC: "1234xy 3425 29.92"
ATC: "Ident received, you are cleared through to KPNE on your current heading at 4000"
En route you may get traffic advisories or other instructions while in the space so you need to stay on the frequency.
Class C (Charlie): For Charlie airspace you only need to establish 2-way radio communication. A response containing your call sign will allow you into the space. Lets look at an example
PIC: "Allentown approach 1234XY 20 miles to the southeast of Queen City at 3000 in bound for landing at Queen City"
ATC: "1234XY standby"
Anytime you hear your call sign, it is considered to be the establishment of 2-way radio communication, if you hear "aircraft calling Allentown, standby", you have not established 2-way communication and must remain out of the space since they did not respond with your call sign. Likewise you may be explicitly instructed that you must remain out of the space.
Class D (Delta): Delta airspace typically extends up to 2500AGL, and you can often climb over it flying VFR. You do not need to hail the tower when doing so but many people do so out of courtesy. If you intend to enter Class D, follow the same procedure as Class C.
You can find the whole breakdown here
Flight Following: Where available and workload permitting you may be able to pick up flight following while in route. You can also coordinate a hand off from them to the airspace you are entering.
First, you should contact the tower before you enter their airspace. Class B (around big international airports) requires explicit clearance to enter, while C/D airspaces around smaller commercial, military and GA airfields require two-way communication with the controller prior to entry (clearance is implied if not expressly denied; you'll hear "remain outside" or "remain clear" if you shouldn't enter).
To initiate communication for any purpose, you usually start by saying the name of the ATC station you are calling, including the type of controller you need to talk to. Let's assume you're flying into Franklin Regional Airport (no clue whether such an airport exists or where it is, just an example). Most Delta towers have a single team of controllers handling every aspect of traffic control, and "Franklin Tower" will suffice. Depending on the airport, they may prefer you use more phase-specific controller names like "ground" (for taxi instructions), "tower" (for takeoff/landing clearance), "departure" (for routing from the runway to the edge of their airspace) or "approach" (for any initial contact where you start outside and want to come in), even if the same small team of controllers in or near the control tower handle all of these things. Since you want to transit, "Franklin Approach" is normally also acceptable. To be friendly (they are human after all) and also to indicate this is an initial transmission with a callsign they don't know about yet, you can start with a salutation, but it's optional.
You then immediately follow this with your callsign (usually the make and tail number of your aircraft, for instance "Piper 342 Lima"), followed by your approximate range and compass radial from the facility, and then your flight mode (VFR, IFR, VFR-On-Top), and finally what you want (in this case to transit their airspace).
So, your initial transmission would go something like this:
Good afternoon Franklin Approach, Piper Three Four Two Lima, five miles northwest, VFR, request transit to the southwest.
The response you'll get depends on several things. Usually, you will hear something along these lines:
Piper Three Four Two Lima, squawk 2451 and read altimeter tape.
This is their acknowledgement that you wish to enter and pass through their space, and for you to do so, they need to be able to identify you on their radar using your transponder, and know what your indicated altitude is (your Mode C transponder will tell them the same, but not all aircraft have Mode C reporting and it can take a few seconds for their display to update). You comply by setting your transponder and replying:
Piper Three Four Two Lima squawking 2451, altitude two thousand five hundred.
The tower's next move will usually then be:
Piper Three Four Two Lima, radar contact, proceed on course.
They are telling you that they can now identify your plane on their screens, and that you are clear to continue flying along your present course at your current altitude. They can, since they control the airspace, instruct you to turn to a different heading, or to climb or descend to a different altitude, in order to accommodate the normal traffic pattern or to route you around another transiting aircraft. If you need to turn, climb or descend without being instructed by the tower, it's usually good to request a heading or altitude change so the tower can make sure you're still going to stay clear of other traffic they control, especially IFR flights.
The tower can respond in other ways to your initial transmission. A critical distinction is that the controller must respond using your callsign in order for two-way communication to be established. If they respond to your transmission in any other way, communications are not established, and therefore you cannot enter their airspace. A common reason they wouldn't acknowledge you by callsign is that your transmission got garbled by other radio traffic or noise interference:
Aircraft calling Franklin Tower, say again.
... or you don't have your mike plugged in:
Aircraft calling, carrier only no modulation.
... or they're too busy to deal with you:
Aircraft calling Franklin Tower, standby.
Also keep in mind that if they heard you loud and clear and respond to you by callsign, they can still keep you out:
Piper Three Four Two Lima, remain clear, expect further instruction.
Even though two-way comms are established, you cannot enter their airspace if they tell you to keep out. If you hear this, their airspace is likely too full to accomodate an additional plane, even transiting. In such a case, if you can get where you're going without crossing their airspace, it's probably a good idea:
Piper Three Four Two Lima copies, cancel request, we will fly around your Delta.
If you can and wish to terminate communications with the tower at this point (since you're going to stay outside their airspace there's not much reason to keep talking to them), you can indicate this by ending your transmission with your callsign instead of beginning with it:
Copy Franklin Tower, cancel request, we will fly around your Delta, good day Piper Three Four Two Lima.
The tower then knows you have likely switched off their frequency and will not respond to further transmissions. If they need to contact you again, they'll likely try on the surrounding MULTICOM or UNICOM frequency, or on GUARD, but as long as you stay outside their airspace there would be little reason for them to try.
If it is just Class D airspace that is your concern, why not just fly over the top without talking to anyone? On a day when a student pilot is flying cross-country, are the cloud bases really likely to be so low that that is not possible? Maybe your instructor was assuming you would do that, hence the lack of instructions on how to talk to the towers.