Since you're interested in designing a mobile processor, yes you should learn more than the basics of Verilog, you're going to have to know it pretty well. Also, be aware that writing functional code in Verilog (this is called front end design) is less than half the story. Turning that code into an implementation (called backend implementation or physical design) is a big job in itself so keep in mind you'll need to learn synthesis, place-and-route, verification, and timing analysis techniques. There are free tools for some of that (e.g. Icarus
http://iverilog.icarus.com/) but they are very old-fashioned in that what you learned will not be terribly applicable to industry.
Since you're just learning, I highly, highly, highly recommend you implement your processor in an FPGA and don't worry about the ASIC part unless you really want to build (it will be extremely expensive). FPGA dev tools are good and orders-of-magnitude cheaper than ASIC tools (since the implementation space is more constrained in an FPGA). I also recommend starting with a GPL'd or public domain processor core (see, for example,
opencores.org) Lattice Semi has a pretty good freeware core available but I forget the name (I haven't used it).
The key to success in IC design is to take baby steps and don't bite off more than you can chew at one time.
If you want to do analog design, start with LTspice
http://www.linear.com/designtools/software/#LTspice and get started with advice and models from this truly outstanding free book:
http://www.designinganalogchips.com/
Don't worry yet about design tools for analog IC design. Eventually you will probably want to use something free like LASI (
http://lasihomesite.com/) or Magic (
http://opencircuitdesign.com/magic/index.html).
If you're ambitious to really design a chip you could try something like this open-source flow from Open Circuit Design
http://opencircuitdesign.com/
Good luck!