I'd second the DE2 suggestion. It's a cheap, but pretty versatile board. It has lots of companion demos and tutorials, as well as a ton of resources online from other starters asking "How the heck do I do <task> on my DE2???" on forums and such.
Quartus II Web Edition is Altera's free IDE, and it comes with pretty much everything you need: a big suite of libraries (most of which are also free to use), a graphical entry environment, an HDL editor, the full synthesis toolchain, and integrated Eclipse tools for writing embedded C/C++/ASM code. Other than that all you'd need to get is ModelSim-altera, the free version of the standard simulation environment.
Altera also has some pretty comprehensive (i) free online training, (ii) IP block, i.e. library, documentation, and (iii) complete reference designs. I'd recommend checking out all three, especially (i), since at the beginning it's easy to get tunnel vision just learning VHDL or whatever, and then realize that you're somewhat clueless as to how to actually get things done in any useful capacity. There's a dizzying amount of jargon and proprietary bullshit, so it's useful to just have someone tell you what everything means, and How It's Done(tm).
All that said, if that ASIC is anything complicated it might be a bit of a big project to jump into. That and it's possible that it would require more resources than an entry level board will supply.
Quartus II Web Edition is Altera's free IDE, and it comes with pretty much everything you need: a big suite of libraries (most of which are also free to use), a graphical entry environment, an HDL editor, the full synthesis toolchain, and integrated Eclipse tools for writing embedded C/C++/ASM code. Other than that all you'd need to get is ModelSim-altera, the free version of the standard simulation environment.
Altera also has some pretty comprehensive (i) free online training, (ii) IP block, i.e. library, documentation, and (iii) complete reference designs. I'd recommend checking out all three, especially (i), since at the beginning it's easy to get tunnel vision just learning VHDL or whatever, and then realize that you're somewhat clueless as to how to actually get things done in any useful capacity. There's a dizzying amount of jargon and proprietary bullshit, so it's useful to just have someone tell you what everything means, and How It's Done(tm).
All that said, if that ASIC is anything complicated it might be a bit of a big project to jump into. That and it's possible that it would require more resources than an entry level board will supply.