Beginner needs help with converting jpg to svg

If you had posted the logo previously I didn't notice the link. After seeing it:

• Live Trace would only be advisable on the citymaps part.
• Draw the green circle using the Ellipse tool and a gradient.
• Downtown can be recreated using the Type tool. You'll need to figure out the font, although I agree with Monika that it looks like Optima.

I don't know how this will be used or what control over the logo design you have, but while you're recreating the logo this may be a good time to think about a redesign/update.

The typeface used in the grey word looks like Optima.

The green thing is just a circle with a gradient (see the documentation about gradients)

The "handwritten" word probably uses also a typeface, but I don't know which one (you could try out www.whatfontis.com). But that one can also be autotraced.

Because a JPEG logo is pixel-based and SVG files are vector-based, you'll need to convert the logo into vectors. You could either do this manually (which is the most work, but would probably yield the best result), or you can use Live Trace to trace it for you. https://helpx.adobe.com/illustrator/using/tracing-artwork-live-trace-or.html 

With Live Trace you may not get a very clean result, especially for a logo and if the starter file is low resolution.