Can’t connect to wireless network at home with OS X 10.5.8

Could be one of many, many things. Have you tried altering the security settings on the router? Are you close to the router? Strong signal strength?

Have you tried connecting other laptops? Is the behaviour different?

Try opening your console (/Applications/Utilities/Console) and connecting. Show the log list and switch to the system log. Watch it, connect to your wireless network, and wait for it to disconnect. Look through the log from the time you connected to the time it disconnected, and see if there are any messages which might indicate what's going on.

Your question is not specific enough, yet, for a proper answer. Give us more to go on.

Even if you get it to connect finally to your network, given the trouble you are having now, you will also probably end up with the wifi drop problem.

There is currently no solution to it other than turning off your wifi and reconnecting or deleting your network settings and reconnecting. Apple has not to date done anything to correct this problem (you can search around and find lots of people complaining about this one).

If you're still in your return period, you might want to think about taking the stinker back and getting a tablet from a company that can at least get the basic telecommunications functions of their tablet right.