You can use iCloud with the phone directly, but iCloud requires a computer with Lion or higher (I won't say "better" since I do not necessarily adhere to that idea for all new system versions).
With your computer you could easily upgrade to Snow Leopard (10.6) or possibly even Lion (might want to get more RAM though). You have to have SL installed in order to install Lion so you might as well get SL and then either go for Lion or wait for Mountain Lion next month (or just stick with SL).
Snow Leopard 10.6 Technical Requirements - http://support.apple.com/kb/SP575
Note, memory/RAM requirements are typically truly minimum and it is recommended you have at least 2x minimum.
Macs and Software that will run with Snow Leopard (Mac OS X 10.6.x) - https://discussions.apple.com/docs/DOC-2455
OS X Lion (10.7) Technical Specifications - http://support.apple.com/kb/SP629
General post on upgrading from Tiger to Lion - https://discussions.apple.com/thread/3920259?answerId=18281569022#18281569022
Lion and Mountain Lion application compatibility - http://roaringapps.com/apps:table
How far you go may be decided by old software you currently run, especially pre-2006. Leopard broke some, Snow Leopard a few more, Lion pretty much the rest.