I would like to ask if is it legal to put someone content on own website (content will be changed to match my own design) without asking the owner of that site for permission. I would like to get some properties available in my area to display on my website from external sources to help people find something to rent/buy in that area. Can I do it without getting agreement with that site?
No, you can't. Property details and photographs are copyrighted, and no one other than the creator can use them without permission. If you just want to use factual data -- asking price, number of bedrooms, etc. -- then it's more of a grey area, and you might be able to get away with it.
Rather than simply putting data on your site - for which you would need permission and to pay money - why don't you develop a site that helps people help themselves?
For example you could include articles on how to search for properties, what to look for in a reputable landlord, links to legal and financial stuff etc. If you do this well enough people might start to pay you to put adverts on your site.
I'm sure if you're intending on having a site that might be helpful to people moving to the area, you could easily get in contact with property management company's and ask their permission to display their items on your website. I don't see how they could not like that free advertisement. But, at least in the US, Copyright Law requires that you gather explicit permission to use something that is copyrighted. Anything creative (code, images, music, etc.) is considered copyrighted the moment it is created.
There are some other things such as fair use that you may be able to get away with. Without knowing the details of your site however, it would be impossible to advise if this would fall under that. I'd recommend you do some googling around for different variations of fair use.