The 180's are a three section boom. The 250's are a four section. That's one of the first identifiers. The upper cab/ engine box style gives you a year range. Anything older has the more rounded upper steel works, anything later is a trapezoid boom.
The model and serial number should have been on a metal chart in the upper cab, behind your right shoulder. If it doesn't have a chart and a model/serial number, it won't pass a inspection, and your going to have a hard time getting a replacement chart for it- without a serial number. Doesn't mean it isn't a bad crane, you just won't be able to use it in any kind of a commercial application that needs annual inspections.
First crane I owned was a tm250.