Which ATC instructions should be repeated back by the pilot? AIM 4-4-7 (b) says:


  Pilots of airborne aircraft should read back those parts of ATC
  clearances and instructions containing altitude assignments, vectors,
  or runway assignments as a means of mutual verification.


As a student pilot soloing into class C airspace, on one flight ATC kept repeating everything back until I spoke it back to him. As I approached the airport he informed me of the bearing (12 o'clock) and distance to the airport until I acknowledged him (comms were pretty busy so I didn't initially repeat back his information).

Since then I've noticed that if I don't repeat back wake turbulence warnings verbatim they will repeat the warning until I do. With a traffic alert I repeat back immediately 'looking for traffic' because if I look first they'll keep repeating the warning.

Aside from AIM 4-4-7 is there any more specific guidance?
ATC is, broadly speaking, required to ensure that pilots hear, understand, and acknowledge information that is delivered.  The way they do that is via read-backs, and although the AIM mentions a few of the things that are required it's a good idea to read back anything that's important (If you're not sure, err on the side of "It's important"), with your aircraft callsign (No callsign? Doesn't count!). 

What are some important things controllers usually expect you to read back?


Clearances (headings, vectors, altitudes, transponder codes, and radio frequencies).
Like the AIM says, mutual verification.
Runway assignments, hold short instructions, line-up-and-wait, and takeoff clearances.  
Taxi routes Instructions, ESPECIALLY runway crossings &amp; hold-short instructions.  
Wake Turbulence advisories.
Traffic point-outs.
The magic words here, by the by, are "Negative Contact" and "Traffic In Sight".
I think if you say "Tally Ho!" you're required to buy everyone on the frequency a bottle of 30-year-old scotch.
Instructions to "Maintain visual separation with" some traffic
ATC is handing off collision avoidance responsibility to you as the pilot.
They need to hear "Will maintain visual separation with" the traffic in question.
They also need you to tell them if you lose visual contact.
Airport Point-Outs (Especially if there are several airports close to each other)
You can also pre-empt this by calling the controller: "Cessna 12345 has Podunk Municipal in sight", possibly with a bearing and approximate distance.
Landing Clearances
If you're in a busy pattern and they give you a sequence ("Number 3, cleared to land") they want to hear that part too.
That altimeter setting you get after a handoff.
(I blame Don Brown for getting me into this habit, but it IS a good one unless the frequency is crazy)


This list is by no means exhaustive, but it's most of the big stuff.