There's a broad spectrum of methods that could be used to monitor your communication.
External Monitoring ("Lawful Intercept")
Your attacker could be monitoring your communication upstream. This could be because they're working with your ISP, or they're sniffing your home network (wired or wireless). You said that your attacker knows things you typed... were you typing these over a clear channel?
Defense against upstream monitoring and local network sniffing: Don't send any communications in the clear. Use only encrypted channels, and use only protocols that provide you assurance that you know who is at the receiving end. The attacker will still see all of your communication, but (assuming the encryption is strong, you don't leak the key, and any passwords are strong) they will be unable to read it.
Defense against local network sniffing: Stop using wireless; disconnect the antenna from your access point, disable wireless in software. For your wired network, inspect your cables. If your cables go through the walls, run new cables between the router and computer that are exposed (visible). You may be able to detect sniffers through various techniques, but it isn't always possible.
Defense against anybody brute-forcing your encrypted communications: (Assuming you're encrypting everything...) Flood the attacker with noise. Through whatever communication channel you're using, make sure that you are always sending something -- random garbage. (It doesn't have to be random, anything will do -- the works of Shakespeare, the text of the tax code, etc.) It should still be encrypted; this makes it so that even if the attacker can feasibly brute-force your communications it is even harder because they won't know which comms are legit and which are noise.
Internal Monitoring (rootkit, keystroke logger)
If your attacker has hacked your computer or phone, there could be a monitoring process that is running and sending all of your keystrokes to the attacker.
Defense against rootkits/loggers: You can run a rootkit detector or anti-malware software to try to find it. If the attack is really clever or uses some new technique, you may not find the logger.
Defense to detect them: They have to get the keystrokes off the machine somehow. Run a sniffer. If you detect outbound traffic that is not yours, investigate it. It may lead you to the process that is sending your keystrokes. As another poster mentioned here, you can jailbreak your phone and run a sniffer to detect any unexplained outbound traffic. This is not foolproof; the logger could be using a covert channel that is not detectable using the common monitoring techniques. Careful analysis of a capture file might reveal a covert channel, but a sophisticated attacker might be using an out-of-band channel or other mechanism that is very hard to detect.
Defense to avoid them on your phone: Ditch the smart phone. Get a "dumb", disposable phone and a prepaid account. Use it for phone calls. Use your (wired) computer (see next item) for Internet commnication.
Defense to avoid them on your computer: Go to the store, buy a USB stick (best if you can find one with a physical read-only switch). From a reliable source, put a bootable copy of your OS of choice on the drive. Switch the drive to read-only. Boot from the drive. (This assumes that you are going to install a hardened copy of the OS onto your read-only drive.) For the most protection, don't access your existing HDD. Wipe it (with every reboot if you're paranoid) and use it as swap/tmp space.
Defense to distinguish between monitoring via rootkit and monitoring via network: Using a clean OS (see above), send a message that you want intercepted from your home network. If the message is intercepted, it is very likely that your network is compromised and not your computer.
Physical Monitoring
If the attacker has placed listening devices or cameras in your home, workplace, or other areas that you commonly use, they could be monitoring you that way. Keep in mind that you could be watched through open windows, or someone could listen through the walls of a neighboring apartment.
Defense to detect this monitoring: (Assuming that you can get feedback on whether particular messages are captured.) Assuming you can surreptitiously disconnect your keyboard from your computer, type something that you want to deliberately be intercepted (in a situation where the keystrokes would not be echoed to the screen). If you find out that the communication was intercepted, you can assume that there's something monitoring your keyboard. Repeat as desired, possibly moving to different rooms. Repeat with the keyboard connected to the computer and characters echoed to the screen, but actual network comms disabled.
If your messages are only intercepted when echoed to the screen, you could be subject to Van Eck phreaking. In this case, you would need to shield your LCD from leaking electromagnetic emissions.
Defense: If you think your home and/or office is bugged, don't communicate from those locations.
Defense: If you think your phone is bugged, get another phone (see above). Never let the phone off your person.
Defense: Hire a specialist to sweep your home for RF bugs. (Or move.) After it is verified clean (or you have moved), apply physical security measures to avoid physical compromise and/or at least have the ability to confidently detect it.
Finally, don't discount the possibility that the person you are communicating with has been compromised. Maybe their computer/phone has been hacked, they are being monitored, or they are otherwise providing with your attacker with info. It's also possible that any servers/services that are relaying messages for you have been compromised and are disclosing your messages.
