Lots of good advice above. Everyone is going to look at their personal experience and give advice based upon what was successful for them with CNC integration and what mistakes they made.
For me it was similar to a lot of what was already said. I started with a low cost evening CAD class at the local tech high school back around 1997. From that I began to get a handle on drawing in AutoCAD and coincidentally met and became fast friends with a local industrial electrician who was making the move to CAD for his industrial electrical drawings.
From there it was my first CNC.....a used Digital Tool router from the late 1990s. It was delivered and assembled over a couple days by the seller who then spent about half a day giving me the basics of Enroute CAM software and the control software, the DOS version of what later became WinCNC......which were created by the fellow from whom I purchased the router. This was back in the beginnings of CNC in the wood shop but there was already a division growing between light duty machines like I bought and heavy iron machines like Motion Master, a couple of the earliest players in wood shop routers.
Everyone has different interests and abilities. Some like to walk up to the CNC, push a button and have it do everything it needs to do. Others like me need to know how things work so take them apart and reassemble them to find out. I took that first 2-head CNC, a Digital Tool 902, and reconfigured it with a home made 4-position tool changer. That was my deep introduction into the mechanical and electrical workings of CNC routers. At least the lighter duty kind.
After the router was in place I was immediately able to run simple parts based upon drawing in AutoCAD and setting up tool paths in Enroute. As others have said that can all be done in one CAD/CAM software package now.....although personally I've yet to see any CAD software that I felt can compare to the ease of use and capabilities of AutoCAD. To this day I'll do simple drawing tasks in Enroute (which for many years has been full CAD/CAM capable) and continue to do most of the heavy work in AutoCAD.
The rest came with just running the system. Over time my AutoCAD, Enroute 3D and G-code editing skills improved. You really don't have to worry about that; the work complexity will increase naturally as you figure out more and more what you can do with the router. It can pay to keep notes though as some of the more complex processes you end up figuring out might not need to be used again for several years.
This all worked out well for me. Now that I'm retired I have another early (perhaps 2000 to 2005) era 4x8 CNC router in my retirement shop that I've spent as much time upgrading to modern standards as I've spent using. I enjoy the upgrade process as much as using the machine. Some other parallel path might be best for you. As I said all any of us can do is tell you what worked for our own businesses.
I do have one small pearl of wisdom though. Don't let your first router bit be an expensive router bit. Start with cheap bits for your learning curve and only after you stop breaking bits move to higher end tooling. I don't really need to say why I know this to be the path to wisdom, do I.........chuckle.
BH Davis