Would it be weird to randomly travel from London to Birmingham ?

It would be illegal during lockdown, but otherwise nothing wrong with going there for a day trip. There is masses to see that it would be worth booking into a B&B and staying a few days.

You must be very young to ask a Q like this. People travel all the time to visit different places.
You really do need to get out more.

Go for it! Birmingham is the 2nd city to London. I’d say the people in Birmingham are more friendly. 

Once lockdown is done go for it.  Random trips are the great.  Maybe stay in a youth hostel for all the random chance conversations to be had?  I used to hostel a lot when I was younger and always ended up cooking meals together with strangers.  We'd put together the odds and ends we'd brought and exchange travel tips and recommendations and I got to hear of so many things I'd otherwise have missed out on that otherwise.  Don't forget to take a tour of the canals.  There are several canal tour operators, take a look on trip advisor.  If you want to go a little further afield you can check out Ironbridge too.

Go for it honestly, nothing random. Quite frankly it's good to explore new places 
There's so much to do! 

Not weird at all. 
Obviously not much is currently open; but in the future, there's no reason not to visit other cities.

You could ask people for approval