This question comes up a lot, and the answers below are compiled from my own experience and a number of posts by others:

http://www.grasshopper3d.com/forum/topics/grasshopper-vs-python-c-v...

http://www.grasshopper3d.com/forum/topics/vb-vs-c-vs-python

http://www.grasshopper3d.com/forum/topics/which-programming-languag...

VB.Net and C#

VB.Net and C# both belong to the ".Net" family of languages, and the things you can do with them in Rhino/Grasshopper are nearly 100% equivalent. Grasshopper itself was written in a combination of VB.Net and C#. Some advantages/comments, in no particular order:

1. Performance - VB.Net and C# scripts tend to execute faster because they are "Just-in-time" compiled as opposed to interpreted. 
2. Autocomplete - both VB.Net and C# have rich autocomplete functionality in their respective script editor components - significantly more so than the python editor. This can be helpful for beginners since you can "hunt" for methods and properties by just typing a "." after an object name and looking at the list of available methods/properties. 
3. Native Component development - If you eventually want to develop GHA assemblies/plug-ins for grasshopper, as of Rhino 5 you will have to use one of these two languages. However, there are plans to introduce python-based plugins in Rhino 6. Even so, the resources around plug-in development are very rich in the C# and VB.Net environments (with c# seeming to be the more popular of the two).
4. "Strong Typing" - VB.net to some degree, and C# especially, are less "forgiving" languages than python - they require you to know about the data type of the objects you're operating on. This can sometimes result in more verbose code - as you explicitly convert from type to type - but it also promotes good programming practice and helps make errors more understandable. 
5. .Net ecosystem - using a .Net language means you have access to the thousands of libraries publicly available, and the process of referencing these libraries and making use of them is comparatively straightforward relative to python. More on this in the following section. 
6. Resources/Support - At least as of 2012, VB and C# turned up more results on this forum than python, and I think you'll find slightly more expert-level coders in those languages able to help you here. 
7. Which one between the two? C# or VB.Net? - Personally, I greatly prefer C# - I find it to be cleaner and clearer to use. I also have some programming background in C++/Java/Processing so I found the "C family" approach to be more familiar. As David and Damian point out in some of the posts linked above, C# is more popular than either python or VB.net in the rest of the coding world. However, if you are learning without any prior programming experience you may find VB.net to be a bit easier to learn. 

Python

Python is, without a doubt, a beautiful and elegant language, which is probably more than can be said for VB.Net/C#. It is very popular with beginner coders, and its syntax is more readily understandable. 

1. Syntax - Python is beautiful to read and write. Its syntax is very clear and free of extraneous punctuation (for example the ";" line endings in c#). It has many very nice language features that make common tasks more concise, like its loop syntax, list comprehensions, list "map" and "filter."
2. Multiple ways to talk to Rhino/Grasshopper - Python enables two general approaches to interacting with the Rhino/Grasshopper environment: RhinoCommon and RhinoScriptSyntax. If you have prior experience with Rhinoscript, you may find RhinoScriptSyntax to be preferable - it adapts many of the methods you're familiar with to the python language, and simplifies some tasks. A word of caution though - working with Rhinoscriptsyntax can introduce a performance hit relative to RhinoCommon operations. C# and VB.net by contrast can only work with RhinoCommon. 
3. "Goodies" - The Python environment in Grasshopper has some "special features" that the other languages lack. In particular, the "GHPythonLib" library enables the ability to call most Grasshopper components from within your code, and the ability to easily enable parallel processing to improve performance. (A word of caution though - these two features do not seem to "play well" with each other, there may be bugs causing memory leaks that result in increasingly worse performance with each execution).
4. Cross-Platform - Unlike C#/VB.net, Python can be used natively in Rhino for Windows and Rhino for Mac.
5. Direct scripting in Rhino - You can also use Python directly in the Rhino environment without the need for Grasshopper if you desire, using the Rhino Python editor.
6. IronPython / Ecosystem issues - one frustration / potential downside to working with Python for Rhino/GH is that though there is a vast, amazing ecosystem of external libraries for Python, getting these to install/work properly in the Rhino/GH environment can be a real pain - largely because the language is actually "IronPython," a version of python designed to work closely with the .Net ecosystem. Many popular libraries like numpy and scipy are very challenging to get working in Rhino/GH.
7. Scripting in other programs - Especially in the AEC industry, Python is a popular scripting language for other applications. Tools like Revit, Dynamo, Blender, and ArcGIS all offer their own Python scripting interface - so learning Python in Rhino/GH can give you a leg up in eventually scripting in these other programs. 
8. Python's Stock is Rising - there are currently a number of efforts to improve the "status" of python within the Rhino/GH ecosystem. The python editor in Rhino 6 has a number of improvements, not least of which is the ability to "compile" add-ons for Grasshopper written in python. I'm sure Giulio can speak to other upcoming improvements. 

I hope this summary helps you find the right option for you. Ultimately you can't go wrong; concepts from any of the available scripting languages will make it much easier to learn the next one. In my day to day work I use a combination of both C# and python, where appropriate, and I love them both. 

I hope others will feel welcome to chime in on this FAQ and add their own thoughts about advantages/disadvantages of these various options! If you have time, read through some of the other posts linked to at the beginning - there's lots of additional great information there.