Motivated by modern applications of light detection and ranging (LIDAR), we study the model of an optical receiver based on an avalanche photo-diode (APD), followed by electronic circuitry for detection of reflected optical signals and estimation of their delay.This model is known to be quite complicated as it consists of at least three different types of noise: thermal noise, shot noise, and multiplicative noise (excess noise) that stems from the random gain associated with the photo-multiplication of the APD. Consequently, the derivation of the optimal likelihood ratio test (LRT) associated with signal detection is a non-trivial task, which has no apparent exact closed--form solution. We consider instead a class of relatively simple detectors, that are based on correlating the noisy received signal with a given deterministic waveform, and our purpose is to characterize the optimal waveform in the sense of the best trade--off between the false-alarm (FA) error exponent and the missed-detection (MD) error exponent. In the same spirit, we also study the problem of estimating the delay on the basis of maximizing the correlation between the received signal and a time-shifted waveform, as a function of this time shift. We characterize the optimal correlator waveform that minimizes the mean square error (MSE) in the regime of high signal-to-noise ratio (SNR). The optimal correlator waveforms for detection and for estimation turn out to be different, but their limiting behavior is the same: when the thermal Gaussian noise is dominant, the optimal correlator waveform becomes proportional to the clean signal, but when the thermal noise is negligible compared to the other noises, then it becomes logarithmic function of the clean signal, as expected.