Run these commands to ensure your repositories are up to date before installing

Copysudo apt update
sudo apt upgrade
sudo apt install python2.7

Then to install pip for python 2:

Copysudo apt install python-pip

To install Python 2.7 you simply need to do the following in Ubuntu 17.10 in a terminal (they work beautifully side by side out of the box):

# refreshing the repositories
sudo apt update
# its wise to keep the system up to date!
# you can skip the following line if you not
# want to update all your software
sudo apt upgrade
# installing python 2.7 and pip for it
sudo apt install python2.7 python-pip
# installing python-pip for 3.6
sudo apt install python3-pip

NOTE: Do not try to remove python 3.6 as it will screw up your system

You can call python pip the following way:

# for python 2.7
pip2 install <package>
# for python 3.6
pip install <package>

Using pip without a number would install python 3.6 packages.

python2.7 version 2.7.18 is available in the Ubuntu 22.04 default repositories. Ubuntu 22.04 is the last LTS version of Ubuntu that contains this package in its default repositories. The phrase "I installed the new Ubuntu 22.04" in your question worries me because the newest Ubuntu LTS is not Ubuntu 22.04; it's Ubuntu 24.04. Please run lsb_release -a to confirm what version of Ubuntu you are using.

The expected results of lsb_release -a are:

No LSB modules are available.
Distributor ID: Ubuntu
Description:    Ubuntu 22.04.5 LTS
Release:    22.04
Codename:   jammy

To install python2.7 in Ubuntu 22.04 open the terminal and type:

sudo apt update  
sudo apt install python2.7  

Ubuntu 24.04

To install the python2.7 package from the Ubuntu 22.04 default repositories in Ubuntu 24.04 open the terminal and type:

sudo apt update
wget http://security.ubuntu.com/ubuntu/pool/universe/p/python2.7/python2.7_2.7.18-13ubuntu1.5_amd64.deb http://security.ubuntu.com/ubuntu/pool/universe/p/python2.7/libpython2.7-stdlib_2.7.18-13ubuntu1.5_amd64.deb http://security.ubuntu.com/ubuntu/pool/universe/p/python2.7/python2.7-minimal_2.7.18-13ubuntu1.5_amd64.deb http://security.ubuntu.com/ubuntu/pool/universe/p/python2.7/libpython2.7-minimal_2.7.18-13ubuntu1.5_amd64.deb  
sudo apt install ./libpython2.7-minimal_2.7.18-13ubuntu1.5_amd64.deb ./libpython2.7-stdlib_2.7.18-13ubuntu1.5_amd64.deb ./python2.7-minimal_2.7.18-13ubuntu1.5_amd64.deb ./python2.7_2.7.18-13ubuntu1.5_amd64.deb

You can still install Python 2.7 from Debian 11. Mixing releases is typically decried as a bad idea, but that is mostly in the context of adding a newer release; adding an older (still-supported) release is much less error-prone, as long as you keep an eye on any packages apt wants to remove.

Add a file, /etc/apt/sources.list.d/bullseye.list, containing

deb http://deb.debian.org/debian bullseye main
deb http://deb.debian.org/debian bullseye-updates main
deb http://security.debian.org bullseye-security main

Run sudo apt update, and you will be able to install python2.7-dev.

As you are probably aware, maintaining Python 2-based projects is becoming increasingly difficult, and the above won’t be a viable approach once Debian 11 is no longer supported (some would argue it already isn’t viable).

First, install some dependencies:

sudo apt-get install build-essential checkinstall
sudo apt-get install libreadline-gplv2-dev libncursesw5-dev libssl-dev libsqlite3-dev tk-dev libgdbm-dev libc6-dev libbz2-dev

Then download using the following command:

version=2.7.13
cd ~/Downloads/
wget https://www.python.org/ftp/python/$version/Python-$version.tgz

Extract and go to the directory:

tar -xvf Python-$version.tgz
cd Python-$version

Now, install using the command you just tried, using checkinstall instead to make it easier to uninstall if needed:

./configure
make
sudo checkinstall

Change version to whichever version you need (version=2.7.1 or version=3.6.0, for example).