I am looking for an existing product out there that would allow me to simply set up and start charging clients for different plan subscriptions.
Ideally, it would have customisable UI and is easy to start.
Take a look at Payhip.
memberstack.io by @DuncanHamra
Isn't Stripe a good one?
https://paddle.com/ (they take care of sales tax too)
chargebee (free until reaching 50k)
Stripe Billing works well for me. If you want a more sophisticated solution go with https://recurly.com/
Not tested yet, but I think I will use this ! : https://tillypay.com/
Are you charging for services or inside a SaaS app?
that would be charging for services
What you can do is on your website have different CTA that go to stripe checkout for each plan.
If you need any help setting it up for free let me know, as I want to integrate it into https://versoly.com/ :)