How to properly charge a clients as a web and mobile app designer?

Interesting idea. I like the Paypal idea.

I think I would charge by the day, in 8 hour chunks.

I am a salaried developer myself. I guess you are a step above me.

Use the Fiverr site. It will let you get customers and they process the payments for you.

It is free to sign up.

I charge by the page. Some pages are long, some are short. I do it that way because it is easy for the client to see and it is easy to count the number of pages to create an estimate.

Things get trickier when you are designing and creating the relational database that supports those pages especially if the client does not specify what he wants the site to do at the outset. I charge a flat fee for the design. I also charge for modifications once the schema has been agreed
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Write a contract. It needs to make the client 100% responsible for content and images. That can protect you from law-suits.

Bill frequently. Does not matter how you are paid as long as you do not release any material from your test site until the cheque clears.

Start on fiverrr...that's where all of you folks end up anyway.

Sounds like you probably code like a beast but didn't take any business classes...

Hourly as the basis usually works fine, as long as your estimates for projects are accurate. If you tell me 5 hours and try to bill me for 10...we have a problem.