Here's a different perspective. I'm a physicist, not a EE, and I have no hands-on experience with antenna design. I did, however, work for almost 10 yrs as a patent agent. I handled applications in a variety of fields, including a slew of applications on antennas and signal processing. Here's my take.
(a) Courses in signal processing won’t prepare you for antenna design. Antenna design is primarily based on the physics of carrier waves, rather than the signals encoded on the carrier waves. You need a strong background in E&M, wave propagation (including reflections and scattering off surfaces), and transmission lines (especially impedance matching). The output of the antenna in receivers is typically fed to a low-noise amplifier (LNA), so you should also learn about LNAs and the proper coupling of an antenna to an LNA. Similarly, learn about transmitters and coupling of antennas to transmitters.
(b) Basic background is covered in physics E&M courses or EE electromagnetics courses. If you wish to pursue R&D in new antenna designs [rather than implementing existing antennas into a system], you will need graduate-level courses, including graduate-level math. The inventors I dealt with all had at least an MS, most had PhDs. Once you have the proper background coursework, there are courses specifically directed at principles of antenna design; just do a Google search on “university courses antenna design”.
(c) There is on-going R&D for miniature antennas for use in mobile devices, such as cellphones and GPS receivers. These antennas are often fabricated using printed-circuit board (PCB) or integrated circuit (IC) techniques, so you should get some background in PCB and IC fabrication. Also E&M properties of materials (taught in physics, EE, or materials science and engineering).