This trophy is for unlocking the ability to build all of the buildings included in the base game (DLC buildings are not required). You also have to build all of the buildings required to unlock the monuments.
Step 1: Progress through the game normally until you reach the final milestone and unlock monuments. This will unlock all of the normal city service buildings.
For the next steps, you will be working on unlocking unique buildings. Based on my experience, I believe that you only have to unlock the buildings listed under the Roman numerals (I - VI). I unlocked this after building the last required building for the fifth monument and I still had several other buildings under the other categories locked (specifically the Eiffel Tower, the casino and the driving range).
Step 2: Unlock all of the unique buildings that require you to do good things. Most are pretty simple, but here are some highlights of ones that take a bit of extra effort:
Statue of Industry
Have 10000 squares of industry
These have to be built squares, not just zoned. Also, doesn't count offices - only the yellow zones. This will take a little while if you focused on offices, but just keep zoning yellow while you play and it will get there eventually.
Official Park
Have 10000 squares of office
Same as above, just with office zones
Business Park
Have 20000 squares of commercial
Same as the last two. I found this to be the most tedious because I didn't have much commerical to start. This needs a lot of residential and a lot of industrial to grow. Keep playing normally and this should eventually happen as long as you keep zoning blue.
Aquarium
Have 5000 children in elementary school
Another tough one. This didn't unlock until I had around 90k population. Just keep building those elementary schools!
Modern Art Museum
Have 50% population highly educated
Just make sure all of your population is close to each of the three school types and keep building high schools and universities until this happens. Libraries also help a little. If all else fails, build the Hadron Collider monument.
Step 3: Make a backup save and then work on unlocking the unique buildings that require you to do bad things. Here are some highlights:
Mall of Moderation
Average Garbage Piling of at least 25
Close all of your dumps/incineration plants and turn off recycling and wait!
Lazaret Plaza
Citizen Health below 20% for 3 weeks
Do this right after the garbage one, since garbage also contributes to low health! Close hospitals and clinics and then move your water pipes down stream of your sewage runoff. Can also build polluting services next to residential zones if needed.
Court House
Crime Rate above 50% for 5 weeks
Close all police buildings and prisons. Can also jack up the taxes for commerical/industrial to increase unemployment rates.
Can also be done as part of the Tough City trophy.
Oppression Office
Unemployment over 50% for 5 weeks
Do this in conjunction with the Court House. Jack up taxes all the way and businesses will close down.
Observatory
Have 1000 abandoned buildings for 5 weeks
This will come naturally with the previous two buildings.
Posh Mall
All tax rates at 4% for 20 weeks
Not hard, but you'll need a decent cash cushion (a few million) to withstand the deficit.
Step 4: Build all the buildings required for each of the monuments. You can click R3 when highlighting the monument in the build menu to see the next building required for each monument.