You'll probably get 1000 miles out of the latest chain, but they do wear faster on a worn cassette. Do plan to replace your cassette the next time you change the chain, though you can try a new chain and see if it skips on your old cassette. You shouldn't need to replace the chainrings, they last much longer than a cassette, typically.
Oil is good, grease is better.
2007 Peter Mooney w/S&S couplers/Terry Butterfly
1993 Bridgestone MB-3/Avocet O2 Air 40W
1980 Columbus Frame with 1970 Campy parts
1954 Raleigh 3-speed/Brooks B72