It depends how old the bike is or how old the tyres are. If there are cracks and signs of ageing on the tyres then replace them but if they look fine and you cant see any cuts, holes or glass then they should be fine. Replace the tube either way!
If your planning on giving the bike a one over yourself then I suggest getting a chain cleaner (or degreaser) from your local bike shop or bunnings and give the chain a good clean up soak it in degreaser (be careful not to get degreaser into your wheel hubs or deraileures) then rub the chain clean with a rag or old towel until it is clean. Good chain lube cant be beaten - Rock n roll gold or Rock n roll red I find to be the best stuff you can buy and most local bike shops sell it.
2 simple steps to get you rolling for your first triathlon! Until you get the bug then its off to the shop to pick up a road bike or tri bike :-)
I would suggest taking the bike into your local bike shop and get them to give it a quick service, they will check your head set, cranks and all moving items are tight and in working order as well as lube your chain and adjust your brakes etc. but that can be a bit costly.
Hope this helps - Good luck and enjoy your first Tri!