In reply to bigrob:
I'll speak for the Slovak side as that's what I'm familiar with as a Central European.
Rysy can be accessed from Strbske Pleso on the tourist path (these are well-marked), which is basically a summer hike with some easy ladders/steps and very straightforward scrambling. This is the only route you're allowed to take there from the Slovak side, unless you're both members of a UIAA club (I believe the BMC qualifies?). Slovakia has a fair amount of red tape within its national parks, hence why the UIAA requirement or needing to hire a qualified mountain guide. That's what you'd need to do for Gerlachovsky Stit, as there aren't any tourist trails going to the top due to it being a more involved ascent (route finding, scrambling, chains, etc).
If you want to do both Rysy and Gerlachovsky Stit then staying around Strbske Pleso would be your best bet. It's rather touristy, but you can find some pleasant traditional villages in any direction if you have a look, at more sensible prices.
As far as flying in goes, that depends on what else you'd like to tick off in one go. But if it's Slovakia and Hungary then Bratislava Airport or Vienna International Airport would both be good options.
Post edited at 17:14