@Linkat,
Your nearest subway station would be on East Broadway, a few blocks away. It can also be reached at Delancey, which is a few more blocks away but a much less gloomy station. (Far less stairs too if memory serves).
Either way, there you get the F train which will take you up to 23rd Street (an actual station).
Another method, seeing as how you'll be right by Pike Street, is the M15 bus. That would give an above ground view as you headed up to First Avenue and 23rd. (Pike Street becomes Allen Street and then at Houston becomes First Avenue) For the return, the bus goes down Second Avenue.
Both deposit you and your kinfolk about 2 crosstown blocks from your destination. One you walk east, the other you head west.
It's not my favorite area (Chinatown), mainly because it's usually congested.
Safety? As safe as any other place. Just be aware of surroundings and know your exact routes. Never look touristy or worried. Think of it as just another city.
You need to get a metrocard to ride subways or the bus. Some/most stations have booth clerks. A few newsstands also have them. A hotel deskclerk might be able to answer and maybe the hotel sells them to the guests.
Remember, to get to Houston Street, it's House-ton, not Hugh-ston)