If you created the logo in Illustrator, it is already in the vector format and should have a transparent background. These things are both baked into a file that you create in Illustrator. You won't need to do any converting. Just save it as an AI file and you should be fine.
Depending on the particular usage of the logo in any particular situation, you will probably have to convert it into various different file types. For Facebook, for example, you will need to create a square image (so if your logo is currently rectangular, or circular, you will need to give it a border so that the final image is square) and then convert it into a file format that Facebook will accept (jpg for sure, and probably other options as well). You will want to reduce it in size and reduce the dpi as well--I usually aim for about 100 dpi for images to be viewed on screen.
First of all, check with your client to see if there are any file types or sizes they want. If they don't know, then you should do a quick review of the collateral where they are using it, so you know the approximate sizes that they use regularly. You want to at least provide them files that they can use that are optimized for use in those sizes. That means at least a set of jpgs, and also some PNGs saved with the transparency.
I usually try to provide a jpg and png that is close in dimensions to the size they will be using on their letterhead, another version that has been optimized for use on the web, and then maybe one more at about 300 dpi and 4 inches on the longest size.
I know that many people dislike the PNG format, but it is the most widely accepted format that includes transparency, so do your client a favor and give them some PNG files.
You will also want to provide your client with the AI files themselves.
If they can provide you with information about the size they will need for the outdoor business signs, then you can provide them with another jpg at that size, but they may not know at this point. Many printers can accept AI files and use those, and if not, and the client does not have the ability to convert it themselves, then I would recommend telling them that they can send you the specifications that they receive from their banner company and you will convert the image if necessary. It is really just a simple set of button clicks, and you will build good will with your client if you offer continued support in this way.