It is very rare to see hardware vulnerabilities that allow someone on the other side of the Internet to exploit you, just through your Internet connection.
It is most common to see hardware issues allow someone with physical access to your machine to exploit your system. (See, e.g., the Firewire DMA holes, cold-boot attacks, and all of the bad things that can happy if someone sticks a USB device into your system.) There have also occasionally been hardware issues that allow local privilege escalation or allow local unprivileged users to crash your machine, though these appear to be rare.
Finally, there are a number of hardware issues that allow an attacker who has compromised your machine (in some other way) to introduce stealthy malware. For instance, some how shown how to hide a rootkit inside your network card or the firmware of other hardware peripherals. In some cases it is also possible to infect the BIOS. These are hard to defend against: you can partially defend against them by using secure boot and trusted computing (e.g., your TPM), but at this stage those defenses are rudimentary and difficult to deploy effectively. The best line of defense is the first one: take other steps to avoid compromise of your machine in the first place.
What should you do? For most people, hardware-related attacks are a much lower risk than other kinds of attacks, so it is not worth worrying specifically about hardware-related attack vectors. You should use standard safe computing practices (turn on automatic updates, back up regularly, use A/V software if on Windows, etc.). If you are concerned, about physical attacks, you could use full-disk encryption: Bitlocker has a good reputation, and has features specifically designed to prevent attacks based upon compromising the BIOS or other hardware elements.