One thing you should take note when getting quotes is that there are many types of quotes. For instance, there are binding and non-binding quotes. So you should make sure that you are getting binding quotes when you request for quotes. This way, you will be sure that there would be no big surprises later on when it's time to pay them for the services they rendered.
To get an accurate quote though, it is advised that you accurately tell them how many boxes will be packed and transported. Otherwise, the quote would only be good for the things you have told them they will handle. It would be best if the removals company would send someone to survey your things. Again, make sure that you show them everything you have to move.
It would better if the quote is detailed as well. At least, you know what you will be paying for. So if there are additional fees later on, you will know if they are reasonable or not. The quote should also provide a ceiling so you will know for sure that your budget will be enough for the fees. So you should look for the phrase that states "not to exceed" followed by the amount when checking out quotes.
And lastly, you should be prepared to make the payments when it is due. You need to clarify when you need to pay the removals company before any service is rendered. This way, you would be ready with your cash and there would be less hassle during the day of the move.