In the end it will depend on where you work. You may not need any programming at all, but this skill is certainly worth having. Generally speaking you can program most things in a version of C or a version of Basic. This goes for small PIC etc. micro-controllers upwards. Assembly language is fine but too tedious and specialised. I think it is good to have tried it, but not to use it, because productivity is low. It is for masochists only.
Embedded and micro-controllers.
DSPs have only C as far as I know. "C" allows you to get closer to the machine, but is more demanding. Basic makes you more dependent on someone else's kernel. All that I tried have been ok with a few idiosyncrasies that you have to work around. There is no particular speed difference, as you expect to use compiled versions of both. I would make sure any compiler has libraries, at least some device drivers and a kernel. If it provides timer functions, and simple I/O for ADC and digital ports and PWM, maybe a RTC, that is all. This implies you will buy a system with hardware on a PCB and a tailored compiler. The PLC is one approach to this that has evolved into a useful tool. I used assembler in the past when memories were small, and Basic with one or two excursions into C for micro-controllers. The kernel may take 100 Kbytes.
At the PC level a visual language is important, as that provides a better user interface. In engineering you will often want to plot graphs, especially the rolling type like a chart recorder (some call it a real time graph). That is something most of the languages are a bit slack on providing, so you have to make your own, and I recommend you make it into a control or class or whatever, so you can just drag and drop to install one or several on a screen object. You will always want serial ports to talk to hardware (USB ports usually use a virtual serial port). You will use ethernet and TCP/IP to talk to WIFI and other computers and websites.
Devices...
Off the shelf PC hardware (like an ADC device) will almost always have drivers that come with it, providing a variety of interfaces from VXD (virtual device drivers that load at boot time), or DLLs. There is a move towards USB to connect such devices now. The driver package is installed, then the software just calls functions.
I was quite happy with VB6 for a PC, but the .NET framework came along and meant relearning chunks of it. VB6 has been crippled for VISTA, and .NET is now VB8 (which you can get free from Microsoft in a level suitable for engineering). Can't complain about that. I have not made much headway with VB8, no incentive. I have to say that PC languages are focused on the user interface, internet, and databases so that engineering applications are not considered much at all. It means you have to think in terms of doing what PCs do.
Open Source? It is getting better all the time, so this means that systems to program for Linux etc are becoming more and more available. There is an advanced version of Quick Basic, but VB like languages are limited. There are versions of C as this is the Linux home ground. There is a commercial version of Basic (Real Basic) that compiles for various operating systems. I have never tried it. I should also mention Power Basic.