Checking in and checking out are the formal procedures carried out on arrival and departure from a hotel. Checking in involves filling in a form and being given a key; notice the use of 'to'. Checking out involves returning the key and paying the bill; notice the use of 'from'. You also check in for a flight at the airport, but there is no checking out. The place where you pay at a supermarket is called a checkout; the verb 'check out' doesn't apply here. The pair of verbs 'check out' and 'check in' are used in computer systems such as document management or version control systems: if I check out a document, I then have the ability to edit it, and nobody else can edit it until I check in my updated version. The verb 'check out' can mean 'investigate' or 'research'; I like to check out the reviews of a film before I go to see it. This is what Cole Daniel was referring to. The verb 'check in' can mean 'report status'; my boss at work likes me to check in at least once a week.

That’s a good question. It’s often said to “check the price” of something; see how much it costs. In your case you can just say “I am just checking the price/I wanted to check the price” Also, another common expression in English is “check it out/check this out”. This is not like checking in/our at a hotel, it means to bring attention to. If I want to show you something on my phone I’d say “check this out” and show you. Or if something is happening nearby us, I would say “check it out” and point in that direction. An even more colloquial use of this word/phrase would be “check it”.