I have only heard of a few viruses for Palms and they were apparently rarely seen by users. Only one I know of did damage to your Palm by erasing the RAM contents. Trend Micro offers an anti-virus program for hand-helds with their PC anti-virus solution.
Spyware COULD attack a Palm, but I haven't heard of any that actually do. It's simply an issue of the size of the user base. There aren't that many people using a Palm online compared to Windows PC's so the hackers who make viruses/spyware don't bother making it for other systems, like Palm's or Apple PC's. This is why Apple PC's are "more secure," not because of any difference in the software or hardware, but because there are simply less users overall to attack, so their devious efforts are wasted on an Apple PC when they are only a few percent of all PC users who will be affected by a new internet worm or whatever. It makes more sense to them to focus on windows users, who are the obvious overwhelming majority.
As far as internet data security on a Palm, it is the same as any other computer. Don't transmit anything on the web you don't want the world to possibly see. Nothing is 100% secure, however SSL is pretty damn good, and again, most hackers wouldn't bother hacking your Palm anyways.
Stories like Paris Hilton having her blackberry hacked, are sort of misleading. For one, the guy who hacked it guessed her password, her dog's name, and din't even access her Blackberry (I am not sure if it's even possible to hack a Blackberry), and he actually logged into her personal web account where her cellular provider keeps an online copy of what is in her blackberry. The lesson isn't that hand-held's computers or cell-phones are insecure, but that you need to choose a good password that no one will ever guess. Use upper and lower-case letters as well as numbers and always use at least 6 characters. Many encryption programs are incredibly secure when you have unique password. Don't use your birthday, or your favourite colour, those words will be checked by any hacker with any real brains. Simple words will be guessed without any difficulty, but if you use even one upper-case character, it will be exponentially more secure.
Bad password: christopher
Good password: Chr15T0ph3r
See the difference?
Still, a person intent on hacking YOUR hand-held computer, as opposed to just any random computer, will be able to if they are determined enough, but I wouldn't really worry too much about it. Most hackers wouldn't even know what they found if they by chance did hack your Palm, but that doesn't mean your security wouldn't still be at risk. If your really worried about it then use an encryption program to safe-guard your most precious data. Use something with strong encryption like 128 bit IDEA for example.
Personally, I don't care if someone hacks into my Palm and steals my contacts, I don't keep anything there important enough to be worried! Anything I don't want to be seen, is in an encrypted notepad program. If someone managed to steal the encrypted data, they would still have one hell of a time guessing my password.
Lastly, if possible, you should have a good firewall on the connection itself, i.e. your broadband DSL internet connection going into a gateway/router with a built-in harware firewall, as well as your firewall software on any PC's. If your accessing the net with your Palm through this connection, it will be fairly safe from any hacking attempts. But remember that any data recieved and sent out onto the web can be intercepted, so I might not do any banking online if your really worried.
Also, any data being sent through a WIFI connection is visible, encrypted or not, to anyone with a receiver in range. They don't have to be able to decrypt it right then and there, they can just record the data you send, and try to decrypt it later. Keep that in mind when using WIFI connections.
Bottom line is that Palm's aren't a very big security risk, because hackers haven't really tried to make them one yet.
I hope I have been at least somewhat helpful.
