First off, welcome to the magical world of Android!
Second, happy new year!
Third, what is your phone? Different phones come with different widgets and apps preloaded.
UCCW is a great tool for both creating your own widgets as well as downloading beautiful templates from Play Store.
If you need weather info, there's also UCCW and Accuweather, HD widgets and many others (you can just search for Weather widget in PS).
Basically, if you want something specific, you just search for it.
(I don't like widgets since they use lots of battery and create wakelocks.)
If you'd like a fresh look, custom launchers are a great choice (Apex and Nova are nice, though there are tons out there).
Rooting the phone will allow you access to your /system partition. In layman terms, you can uninstall preloaded apps, make various optimizations, do advanced backups and more (CPU UC and governor change). (Also hacks in some games with a special app
)
If you really want a new look, you can flash a custom ROM. Cyanogenmod is really popular, though Paranoid Android is good if your phone's screen is big (Note 3, and really anything over 5 inches). There's also MIUI and various variances of your stock ROM.
If you want to improve performance, you can flash a custom kernel. Those usually bring optimizations and CPU and GPU overclock, all of which boosts everything (you can also save power by underclocking and undervolting).
XDA in general is a great source of info as well.
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