An extremely vague question that will get just as many different answers as users posting. Everyone gets to market a bit differently.
I am a software engineer, so my journey through the electronics world will probably be similar to yours. Fortunately, I have the privilege of working with some extremely knowledgable double E's.
What I typically use Arduino for is quick-and-dirty proof of concept. I usually code directly on an AVR for serious projects. So your understanding of what Arduino is, I would say is accurate.
First thing that I started with was some solderless breadboards - http://www.ebay.com/itm/390339573267?ssPageName=STRK:MEWNX:IT&_trksid=p3984.m1439.l2649#ht_1198wt_967
Get a bunch of those, add a nice variety of resistors, LEDs, etc.. Learn circuitry and theory, move on to ICs (555s Op-Amps, etc..)
Once you have a circuit that you like, solder it to some perfboard. perfboard for sale | eBay
The entire time, keep an up to date schematic of your design. It helps with learning theory, and for posting to forums for help. No doubt you will let the magic smoke out of a few components and will need to know why.
Download Eagle. Better yet, if you know someone knowledgeable with a PCB CAD solution, take their advice. (You'll ask for their their help later, no doubt) Or, if you are not planning (or interested in) on learning PCB CAD - have them create your PCB layout for you.
Depending on your PCB design, the number of layers, complexity, etc.. And how often you plan to do this sort of thing, and what kind of physical area you have available to you. You may want to create your PCB at your home, or have it shipped out to a proto-board house. But - before you pay any cash to make your PCB, I strongly suggest posting your files to a forum to have them looked over. Sparkfun.com is a great place for that.
When it comes time to send your board layout out. I would ask around in your local area to see what everyone there is using. If you, or anyone you know work for a manufacturing company that designs PCBs for their products, ask them who they use. They will usually have good reasons on why.
Depending on what you are doing, you may not ever venture further than the perfboard step. I have seen some cool gadgets running entirely on perfboard.
Keep everyone posted on the awesomeness that you create. Have fun and good luck!