Hi @PrDr
For the most part, everyone that has posted so far has provided great suggestions. But what I sense is that you have an anxiety or fear of rejection from approaching people.
Either way, I have a different approach that may appeal to you:
Next time you see someone walking towards you, try to identify something that they made a conscious choice of and make a compliment.
Examples:
“Nice shirt. Looks good on you.”
(anyone with a distinctive shirt)
“I love your hair”
(anyone who appears to have put effort into their hairstyle)
“Cute dog” (person walking a dog)
They will likely reply with a “thank you”. You simply say “you’re welcome” and proceed walking in the opposite direction of them.
Some people may not reply and just make eye contact and either smile or give a head knod. Consider this equivalent to the statement above.
Others may not make any eye contact at all and proceed walking as if you weren’t there. They probably didn’t hear you or they had must have had earbuds in.
The whole idea behind this approach is you have a time limit and a set script. If anyone is caught off guard it will be the other person, and if so they will likely appreciate the compliment, so don’t feel bad. This will get you used to making the first move in a conversation.
After you become comfortable with doing this exercise, if they verbally reply with a “thank you”, stop or slow your pace and gradually follow up with a question:
Examples:
“Where did you get it [the shirt] from?”
“How do you get it [their hair] to look that way?”
“What kind of dog are they?”
When they reply to your question, you can ask additional questions or compliment again and proceed walking.
Examples:
“That’s cool. I’ll have to check that place out.”
(in response to the shirt)
“Interesting. It [their hair] looks great on you.”
“My cousin has the same type of dog. They’re great dogs.”
Even if the topic is not something they made a conscious effort in, people generally enjoy talking about themselves. The secret to make things flow well is to sound genuine, act like you’re actually interested in their answer.
I hope this helps you and anyone else that may have trouble approaching strangers.