I usually get emails from people that have recently switched over to android asking what are some of my favorite widgets/apps that I think they should download. Here is a short list to get you started off with customizing your phone:
1. First and foremost you need a launcher so you can change your dock, resize widgets, etc. Launcher Pro Plus and ADW Ex are two of the most popular.
2. My FAVORITE clock/battery/date widget is Minimalistic text which i did a really basic tutorial on. You can make separate widgets with this and layer them how you like.
3. I don't like a lot of icons on my screen so most of the time i use a folder widget like SimiFolder or Circle Launcher with a transparent icon. You can use any icon you like.
4. A popular easy text clock/date widget is Clockr Evolution.
5. Beautiful widgets and Fancy Widgets Pro for weather, you can skin these and they also have the large clocks that you see in Sense or from "Android-like" blackberry themes.
6. Pure Calendar is my favorite calendar widget as well as smooth calendar. Smooth calendar just shows three entries at a time. Pure calendar has tons of options.
7. A RINGTONE APP! Lol. I use "Music Ringtone" from the market. Yes its called that. It's icon is an old sheet of music with red notes if that helps. You can download full songs and crop them into ringtones.
8. If you use twitter my favorite right now is Twidroyd (or however you spell that damn app) because you can download themes for it that other peopl have created. Plume has too many refreshing issues for my tastes and a few other people have had the same problem but its still worth checking out. It was my favorite when i had a G2.
9. I have a wallpaper app in my phone for when i get bored. Flikie has a pretty decent selection. If you like girly patterned walls there is one called Pattern Wallpapers that has thousands.
10. I almost forgot! Desktop Visualizer lets you resize icons to how ever big you would like. Once you install it, you add it like any other widget. Itll place it on the screen, tap it, and choose an icon from your gallery that you have already downloaded to your phone and then choose what app you would like it to launch. If you are using Launcher Pro or ADW Ex (which i hope you are!) you can longpress the widget when you are done and resize it. (For launcher Pro you just longpress it and release it in the same spot to make the resize box show up.)
I might add more later but these are my must haves. Hope this helps!