If you're not already connected to Wi-Fi, you'll need to connect now. See How to Get Internet on the iPod Touch if you need help getting online. It's the white icon with a blue compass inside. You'll usually find it on the home screen. This opens your web browser. It's the wide gray bar at the top of the screen. This springs open the keyboard. The Search key is at the bottom-right corner of the keyboard. This opens the mobile web version of YouTube. To browse, swipe down and check out recommended videos, or tap the flame icon at the top of the page (in the red bar) to see what's trending. To search, tap the magnifying glass icon near the top-right corner of the screen, type what you're looking for, and then tap the Search key. The video will start to play. The free version of YouTube is ad-supported, which means you may have to watch an ad before the video will play. Sometimes you'll have the option to skip the ad by tapping a Skip Ad button at the bottom of the video.

Summary:
Connect your iPod Touch to the internet. Open Safari. Tap the address bar. Type youtube.com into the address bar and tap the Search key. Browse or search for a video. Tap the video you want to watch.