In this paper we are interested to learn from a wireless digitally modulated signal the number of antennas that the transmitter (Tx) of this signal uses, as well as its specific modulation scheme (from phase-shift keying (PSK) or quadrature amplitude modulation (QAM)). Formally, these are modulation and antenna classification problems. We examine the problems with data-driven machine learning (ML)-based techniques. The two sub-problems of modulation and number of transmitter antenna classification are initially examined independently for a variety for system parameters, namely the SNR, number of receiver (Rx) antennas, and classification algorithms. Then we consider the joint problem where we follow two approaches. One, where the sub-problems are solved independently and in parallel, and one where the antenna classifier waits on the result of the modulation classifier. The two proposed schemes do not require any knowledge/details of the used modulation schemes and the way the Tx antennas are used (spatial multiplexing, space-time codes,etc.) as it is fully data-driven and not decision-theoretic based. The results of our approach are characterized by high classification accuracy and they pave the way for more ML-based data-driven techniques that reveal more characteristics of the Tx.