I want to improve my social skills. I thought by way of stepping out of my comfort zone. So here’s what I thought:
Try and make conversation to a random person, location or occasion be damned, at least once per week. With saying more than two sentences be deemed sufficient. If the other person does not initially respond amicably (to the first sentence), I leave them alone.


I think you are headed in the right direction, and if you are excited to execute this plan then you should do it.
However, if you find that you aren’t sticking to your plan, I suggest some modifications.
First, increase frequency to multiple times per day. To really make progress in your social skills, you need to build the habit and build up a lot of volume.
Second, decrease the difficulty. “Really pushing your comfort zone” feels like progress, because you have to push hard and you feel like you tried hard and were courageous. And I want to emphasize again - if you are consistently excited to go out and do this and you are meeting your goal, great! Keep doing what you are excited about. But for most people, this will end up like running a marathon by breaking it up into sprints. You try really hard and make some progress really fast, but then you get tired and burnt out and have to take a break before your next sprint. Most people, most of the time, find they make better progress in long term goals by applying a small, continuous amount of effort - running the marathon at a consistent, light jog.
So instead of making conversation with a random person in a random location, you could say “this week, I will have 100 conversations. Talking to the same person multiple times counts. Talking to friends counts. The interaction I have in the checkout line counts.” Then you can look back at your week and say “wow, I really did talk to a lot of people, and none of the conversations were hard”. And since you know that having the ability to strike up a conversation with a stranger is your eventual goal, but isn’t an absolutely-must goal, you might simply see the opportunity to talk to a random stranger and take it, and find it to be easy in that particular moment as well. When working with things like social anxiety, taking the pressure off is often the key to making quick progress. Then, the next week, you can challenge yourself “I will talk to 100 people I’ve never talked to before.” Then the next week “I will talk to 100 people I’ve never talked to before - service workers in their normal role at work don’t count.” Etc. Each week, you expand your comfort zone little by little, with every rep feeling easy peasy.