Q:

Bash does not remember commands

It's well known that if you push «up» and «down» keys in gnome-terminal, you can move on used commands. It only works for the commands wich are introduced before closing the window. When I close it, gnome-terminal forget all commands.
Also, I cannot paste text w/ Ctrl+Shift+V.

A:

The quick answer is 'add the following to the very end of your .bashrc file': 

shopt -s histappend
export HISTSIZE='10000'
export HISTFILE="$HOME/.bash_history"
export HISTIGNORE=cd:ls:bg:fg:exit
export HISTCONTROL=ignoreboth:erasedups

...Or something like it, that's directly out of my .bashrc file.
The correct answer is 'read up on the bash shell'.
I highly recommend Learning The Bash Shell (http://oreilly.com/catalog/9781565923478).
Otherwise there's a lot of free material on the web to get you started, for example : 
http:www.ibm.com/developerworks/library/l-bash.html
http://tldp.org/LDP/abs/html/
Also, Romanovzky and Dennis have a valid point - the file may already be there.
You might also make sure your user owns and can write to your .bash_history file : 

sudo chown $USER:$USER $HOME/.bash_history
sudo chmod u+w $HOME/.bash_history

