I'm a Warwickshire birder and I love Minsmere (and Titchwell, the Farne Islands and any of the other famous hotspots I've visited in the last three years of so).
But...
I get just as much pleasure, if not more, from my patch, and by that I mean the area around my house as well as the county's better known reserves (Brandon, Draycote, Ladywalk etc). Last weekend for example, I found a Common Sandpiper less that a mile from my house, a patch first for me. Then the next day I turned up at Brandon to find two Temminck's Stints there. I got far more excitement from the Common Sandpiper than I did from the much rarer stints. Not only was it was a first for me on my local patch walk (indeed the first wader) but it was self found.
Many of my favourite birding moments have been with a couple of miles of my house - a Spotted Flycatcher, Little Owls, a breeding pair of Tree Sparrow, Goosander, the sandpiper etc. Not as rare as some of the birds I've seen elsewhere, but I've found all of them myself, they've all be unexpected and no one told me they were there or posted them on a whiteboard / photocopied sheet. That adds something special to the experience.
Don't give up on your patch wherever you live. I've been lucky enough to spend a little time with a couple of experienced county watchers and you'd be amazed what is on your doorstep (I mean seriously amazed - I've found out that several of the birds on my wish list are regularly(ish) found within 10 miles of my house!).
I think the only difference from Minsmere etc is that it takes a little more time and experience to find those special birds on your patch (unless you're looking for Crested Tits, which could be a little harder), but when you do, it is well worth it.