question related to transferring site to other server

I have an article directory, it has around 150000 articles posted.

My current host refused to host in the plan which I had taken but there is too much space availabe.
They are saying that you have to go for VPS, but I'm not willing to pay him around $30 per month.

I found one new hosting company that is bighost dot com, it said that in $10/year you can host one site unlimited space.

I signed up it but while transferring back up file he refused that we can't transfer your file you have to do it manually as it has any apache file which is causing our server.

I chatted with their executives many time, they uploaded files many time and they didn't finalized later a supervisor said that you have to yourself.

How can I do, I think they are not willing to host so big site or what.

If I want to transfer myself then what knowledge should I have?