When did we first observe annihilation of antimatter? Firstly, when did we actually observe this process? (I assume we have but I may be wrong about that as well.)
Secondly, did we expect it to happen, or not?
 A: Positrons are the antiparticle of electrons:

Dmitri Skobeltsyn first observed the positron in 1929. While using a Wilson cloud chamber to try to detect gamma radiation in cosmic rays, Skobeltsyn detected particles that acted like electrons but curved in the opposite direction in an applied magnetic field. 

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Carl David Anderson discovered the positron on 2 August 1932, for which he won the Nobel Prize for Physics in 1936.Anderson did not coin the term positron, but allowed it at the suggestion of the Physical Review journal editor to whom he submitted his discovery paper in late 1932. The positron was the first evidence of antimatter and was discovered when Anderson allowed cosmic rays to pass through a cloud chamber and a lead plate. A magnet surrounded this apparatus, causing particles to bend in different directions based on their electric charge. The ion trail left by each positron appeared on the photographic plate with a curvature matching the mass-to-charge ratio of an electron, but in a direction that showed its charge was positive.

It took longer to find the antiproton:

The antiproton was firstly observed in 1955 in Berkeley, by Owen Cham-berlain, Emilio Segre, Clyde Wiegand and Thomas Ypsilantis (for this discovery Chamberlain and Segre won the 1959 Nobel Prize in Physics). They grounded the experiment also on the informations obtained by the tests with cosmic rays.

Since the Dirac equation , used for calculations in particle physics, allowed for opposite charge particles, antiparticles were more or less expected/sought for.
The positron annihilation was studied in the1940's. For antiprotons it was part of the discovery .

It involved photographic emulsion stacks to pick up the telltale star-shaped bursts indicative of an annihilation event from a proton-antiproton pair.

