The most interesting thing to see in Odessa is the old town itself. The city was once the center for trade for the Russian Empire as well as an intellectual and artistic centre prior to the revolution and during the Soviet Union. Much of the grandeur of the city dates from the period before the Soviet takeover and subsequently Odessa shows its age.
The economic hardships that befell the city falling the collapse of the Soviet Union in 1992 have left vast portions of what was a magnificently wealthy old city in a state of total disrepair. The old city though is quite clean and feels very safe so it makes for a good two days worth of casual unguided wandering particularly with the wide tree lined avenues and large open parks.
In the much smaller and better kept part of the old town there is a large and beautiful Opera house and some very nice parks. There is also one main street leading through the centre that is vibrant with people selling street goods to tourists.
If you're looking for a nice route in a city center, try go from Grecheskaya square through Gavannaya st., then onto Gogolya street, in the end of which turn right and you will see Tyoschin bridge Mother-in-law bridge. Walk through the bridge and take a stroll along the Primorskiy boulevard. In the end of the boulevard you'll see the city hall. Turn right and go up to the Opera House, from where you can get to Deribasovskaya street. It's especially beautiful in the evening.
Also there are many interesting museums in Odessa.