How did the performance of jets in World War II compare to the cutting edge propeller planes (e.g. P-51, Zero's, etc) of the day?
Much longer take-off distance and less endurance, but higher speed at all altitudes with jets. The performance advantage of jets widened with increasing altitude and resulted in a higher service ceiling. Both the Me-262 and the Ar-234 would fly at least 100 km/h faster than any propeller airplane, and the Ar-234 could fly reconnaissance missions over France and England in complete impunity. A Me-262 test variant (262 V9) with a low-profile canopy reached 600 MPH in level flight.
This gave jets the possibility to break off combat at will. They could only be engaged with success when they were flying in for a landing with empty tanks or by chance, when the jet pilot was unaware of his opponent. Since the Me-262 was designed to intercept bomber formations, it was not suited for dogfights. The Heinkel He-280, on the other hand, was evaluated in 1943 against a FW-190 and would easily outturn the propeller aircraft, so this was not an engine-related characteristic.
Note that the first kill of a Me-262 was by a de Havilland Mosquito, a very fast propeller aircraft that could outrun almost all other propeller aircraft of its time.
I don't have much in the way of numbers, but, at least from reading the autobiography of Chuck Yeager ("Yeager: An Autobiography" - http://www.amazon.com/Yeager-Autobiography-Chuck/dp/0553256742) that he shot down an Me 262 (one of Germany's jet fighters) by exploiting a weakness of the aircraft (such as their long landing approaches at relatively slow speeds). He notes that, as @Peter Kampf says, that it was really hard to engage 262s due to their extremely high speed, but also keep in mind that this doesn't make them a good fighter -- especially not against slower aircraft when both only have guns. If the aircraft was designed to cruise at high speeds, it would be extremely difficult for it to maneuver at low speeds due to the higher wing loading (e.g., maneuvering loads become higher for a lower airspeed and you become structurally limited on how tight your turns can be). Hence, a slower aircraft could, potentially, fly circles around such a plane. If it could convince the jet to slow down in the first place...
However, because of their high speed, Yeager notes that they could dive through a bomber formation with relative impunity. They would not have long to fire on the formation, but it was incredibly difficult to hit the jet fighter.
Anyhow, according to the National Air and Space Museum's website (http://airandspace.si.edu/collections/artifact.cfm?object=nasm_A19600328000), the Me-262 had a top speed of 540 mph (unclear if that is in level flight, but I'd believe so). Looking at a max. level flight speed versus altitude for a P-51B, it never breaks 450 mph (source: http://www.wwiiaircraftperformance.org/mustang/p-51b-24771-level-blue.jpg). This performance plot is from an Army performance test in 1943 (source: http://www.wwiiaircraftperformance.org/mustang/P-51B_Official_Performance_Figures_15May1943.jpg).
The speed advantage of the Me262 gave it an advantage in combat over the piston engined aircraft used by the Allies, once German pilots had developed suitable tactics which exploited this advantage while minimising disadvantages such as the engines' slow throttle response, and the need to operate the throttles slowly to avoid engine failures and flameouts. The Germans found that the best approach to flying the Me262 in combat was to leave the throttles alone as far as possible and use climb and dive tactics. When flown in this way the Me262 was effective against both fighters and bomber aircraft. Pilots learned to avoid turning dogfights with the more manoeuverable Allied fighters. While the explosive and incendiary shells fired by the Rheinmetall MK108 30mm cannon were very destructive, pilots complained that the relatively low 540 m/s muzzle velocity was a handicap in fighter versus fighter combat and also in ground attack work.
One other factor to consider... while the ME262 had a higher top speed, it would tend to lose a lot of that speed in a tight turn. That, combined with its slower acceleration meant that traditional turning/dogfighting tactics were not the optimal for the jet fighters.
The best ME262 pilots tended to use the same basic tactics that US pilots used in the Pacific against the slower but more nimble Japanese fighters: diving attacks at a high speed the opponent couldn't match, then pulling up to trade speed for altitude at a rate the opponent couldn't match, then reposition and repeat.