Actually i can suggest a few places to visit... First of all you want to see...Wait i will make a list for the places to visit :

These places which i mention are in the city centre...They are all generally historical places....i attach google maps links also...

1. Smyrina - Agora : https://goo.gl/maps/QRpBFiwXcQp

2. Kapalı Çarşı - Old Bazaar : https://goo.gl/maps/uHa4s8vcemN2

3. KadıfeKale : Kadıfe Fort : https://goo.gl/maps/RYXyBkaYnDv ( you need to be careful in there because of it is a little bit dangerous place.Actually i want to say there are bad people who want to steal your camera or phone and maybe your money...So be careful. )

4. the historical clock tower : https://goo.gl/maps/efNcJvqLKmq it is in the Konak Square

5. Saint Polycarpe Kilisesi - Saint Polycarpe Church : https://goo.gl/maps/ErnbLR76QaS2

6. Historical Elevator : https://goo.gl/maps/qckSf7dYDM62

Now a few far away destinations :

1. Ephesus : https://goo.gl/maps/U2mjvvMq9m52 You can visit a lot of places in here and from there you can go to the sea for swimming to Kuşadasi and also you can visit Kuşadası Castle.

2. Kuşadası Castle : https://goo.gl/maps/Afnt247t5S42

3. Şirince Village : https://goo.gl/maps/NJEUQ6bCNrE2 really beautiful village with its historical houses and wines

PS : For now it is all i remember and if you want i can join you for the closer places to the city centre... :)