I am not a lawyer and so my advice will be very limited and will be based on common sense. Before you have taken the trouble to create an exchange you should have researched the laws of your country, for this you could count on the help of a lawyer in your country. But leaving that aside let's your question:
I know that until sometime ago most of the exchanges and trading platform didn't had any license, is there a way to get started without a license (legally) and apply for it later on?
as far as I know, in the United States, Japan and South Korea are the countries that require licensing and they are very strict, and the most rigorous are Japan and South Korea. in malta you can also have a license, but from what I read they are more considerate. If you are in Russia, Saudi Arabia, Brazil, Portugal, Italy, Argentina, Canada and many other countries you may not need a license, see the case of yobit, hitbtc and many exchanges that have no license, but are online and making profits by doing poor service and robbing customers. Of course, the fact that you do not have a license and not be inspected by the government does not mean that you should be a scammer who just wants to create an exchange to enrich by stealing your customers' money
I wonder how most of cryptocurrency projects can afford to go live if they have low budget like myself.
Oh, you do not know. today people have discovered the gold mine to finance their projects that 90% of them are just meaningless projects, I'm talking about ICO. ICO is the new way people found to get funds for their projects.