I'm like you; I was also born the year before Commodore filed bankruptcy (1993). At this point, I have an A1200 with a modest 4MB expansion, an A1000 with 512K chip, a PowerPC Mac G4 with MorphOS, an A500 on the way (perhaps), and a system with AROS, as well as UAE on my main PC.
The world of Amiga is awesome. Just say what you want to do, and people here will be able to suggest all sorts of software and hardware combinations that will allow you to do it.
If you want a classic Amiga experience, get an A500 with a bunch of disks or an A1200 with FastRAM and a hard drive to run WHDLoad on. There's also something new called the Minimig which is an implementation of an ECS A500 on a small FPGA board.
If you want to go modern, go with an Amiga with a PowerPC accelerator (not cheap), a PPC Mac with MorphOS installed, or x86 with AROS. One could also wait around for the NatAmi or AmigaOne machines, but those have been in development for a very long time and will probably be rather expensive at launch (especially the AmigaOne).
Of course, there are emulation options. WinUAE is always good, especially with AmigaForever. X-Amiga is a special distribution of Linux made for use with UAE, and Amithlon, despite being inactive for several years, is still pretty good if one's hardware can run it.