There are lot of plugins available in the market for Weather and calendar that you can use in your Visual web part. Also there are some articles you can find here to how to create weather web part in SharePoint.
World Clock and Weather Web Part - SP2013.
If you know how to use jQuery plugins then you can simply refer that files and you will get your result in Visual web part. Also see this example
Weather web part. Similarly you can do for calendar web part.
Refer this
How to create Visual Web part.
Hope this helps!