There are hundreds of species, similar ones often distinguished by details of parts such as fruits, flowers and leaves. Looking at internet photos alone may not produce the same species, or result in a false identification - unless species of interest is very distinctive and commonly planted.

Comparing parts of tree being wondered about with those in a nearby labeled collection might turn up a useful lead. Always follow up with reading serious written descriptions.

If you live on the east coast, it is likely one of the E. deanei/ E. saligna/ E. grandis trio. More likely one of the latter two.
They all have tall, straight, white, shaft-like trunks which soar into the stratosphere (or so it seems) . E. saligna has a bit more rough bark around the base than E. grandis. They both favour the sub-tropical coastal areas N of Sydney, while E. deanei grows in the Blue Mountains.

Note the bangalow palms: this area is 250 km N of Sydney and hence has a subtropical climate.

{{gwi:496218}}

When you mentioned colored bark, the Rainbow Eucalyptus came to mind. This tree has the best looking bark of any tree I've ever seen.
http://www.terragalleria.com/pacific/maui/maui-north/picture.hawa2592.html

It doesn't fit other parts of your description though, but I thought others would enjoy seeing this tree.
BoTann

Here is a link that might be useful: Rainbow Eucalyptus