This is a duplicate question, although this is actually asked much better than the other one is.
Generally, your picture of what goes on is pretty good.
As mentioned in a comment, there is Clearance Delivery, which issues your ATC (IFR) clearance. Except... at most of the larger airports, that function is now automated and the clearance is passed via ACARS, which is essentially a closed e-mail system direct to the aircraft. So it's actually common to NOT talk to Clearance Delivery now in airline operations. You still do at smaller airports that don't have the PDC (Pre-Departure Clearance -- the automation that sends the clearance itself over the ACARS network) installed, and you'll call Clearance if you have a question about how you were cleared, and in a few other uncommon situations such as a reroute that's issued after you got your initial clearance.
Ramp Control exists at some airports, and it's essentially a "ground control" function for a separate, isolated area. So in Atlanta, you actually have several of these, one for Ramp 1, which is the east side of the A terminal and the west side of the B terminal, another for Ramp 2 which is the west B gates and the east C gates, and so on. They control the "in's" and "out's" of their respective ramps, including coordinating who will push back when. As the aircraft leaves the Ramp Control area, it's handed off to Ground Control, and things proceed normally. NOT all airports have this, and not necessarily every terminal (or gate) at a given airport will be controlled by Ramp Control. Gates that aren't controlled by Ramp will typically call Ground to push. SFO is like this.
Typically, controllers see most of what you've filed on your flight plan, so they know before a given flight is passed to them that SkyPig 23 is a 757 that departed JFK, is filed for FL 360, and is going to ATL with a routing of whatever set of airways, points, and STAR. NOT every controller will see everything in the flightplan; I think the TRACON sees remarks but the Center may not, was one example I vaguely remember from a tour a while back. But essentially they know who you are & what you're doing by the time you switch to their frequency. Sometimes things get confused or the pilot requests a change (in altitude, routing, or whatever), and sometimes the controller has to change the plan up, but generally you check in at whatever altitude (and climbing/descending to whatever, if that's the case) and the controller acknowledges and that's that until he has a new clearance (further climb or descent, new routing, approach clearance, etc) to issue.
Mostly now, airlines are filed on both a SID (standard instrument departure) and a STAR (standard terminal arrival route) so "requesting vectors" in pretty uncommon. Could happen if your FMC (flight management computer) dumps your routing, but that's rare, and usually operator error when it does happen. A departure or an arrival might INCLUDE vectors (i.e. fly heading 350, climb & maintain 5000', expect radar vectors to JEBBB), but in those cases it's expected.
Multiple approach controllers are pretty common, especially in the larger terminal areas. Often you'll be switched from Center to one Approach Control frequency, who may be handling all arrivals from the east, with another handling everything from the west. Then as you get close to the airport, they hand you off to "Final", which is still "XYZ Approach" but his job is sequencing everybody onto the base, dogleg, and final, clearing you for the approach, and then handing you off to tower. If the first Approach controller owns airport to 30NM out surface to 17,000', the "Final" controller might own 7000' and below within 12 miles, and Tower owns 3000' and 5 miles. Serious generalization there, but that's the idea... each successive layer of the onion owns a smaller but busier piece of airspace. Letters of Agreement between the Center and the TRACON and then procedures within the TRACON will spell things out in great detail.
Hope that's helpful. Good question asked well; hopefully this won't be lost to memory just because it was asked 2nd.