Hi Mike, welcome to the forum! To try and answer your questions directly:

- What is the best weight class to start with? I would assume either beetleweight or antweight but I am not sure
This depends on what you feel comfortable with. For a completely new beginner, I'd recommend either the Beetleweight or Featherweight classes simply because of the availability of the parts and the scale of the parts. Antweights is another option but they can be, in my experience anyway, fiddly.

The flipside is that AWs have quite a range of events throughout the year (Ant Freeze, AWS, BotFest and so on though disclaimer, I run the BotFest events :P). BWs tend to be a little few and far between but they are growing. FW's tend to share with the larger events like Extreme Robots and Robots Live. There is also the RoboDojo events in Leeds which is a FW only event with a beginner friendly sportsman level.

- What is the simplest weapon to install to make sure that an active weapon system works and I can actually program it?
In my experience, a lifter is perhaps one of the simplist while teaching the basics of active weaponry with axes next. You might imagine a saw or something would be easy but I wouldn't ever recommend these to a beginner because of the capacity for things to go diasterously wrong.

- Where would be an ideal place to source materials from (transmitters, motors etc.)
HobbyKing is generally a good place to start for RCs. Motors, depends on the weight class but FWs tend use drill motors as a starting motor. BWs, try here: https://shop.bristolbotbuilders.com/

- What is a usual budget for building a machine?
This is one of those questions that is quite subjective personally. If I had to price up one of my previous FWs as a "basic" machine from scratch... about £350 all in. One of my BWs... about £150 all in (so that is motors, wheels, shell, fixtures, ESCs, wiring, RC control). This does, however, greatly depend on what specific parts you get.