Vectorise logo

Hi everyone.

I’ve made an attempt at vectorising my father’s company logo, so that I can use it to create summer postcards for his clients and more. Unfortunately, the logo I have access to is not very high res, but it will have to do. I would appreciate if anyone skilled at vector art and illustrator could please help me improve my vectorisation and also how I can add the colour to my vector logo to create a complete replicate of the original. I really want to improve my skills, so if you could tell me what you did, it would be very helpful.

Cheers

P.S. I’ll have to send the zip file to whoever wants to help me out because it’s too big to upload here!