I'll be taking my first international trip later this year and I've found that the US consulate just a few blocks from where I will be staying. I'm curious about what goes on at the consulate, how it operates, and I kind of just want to see it. Are consulates the kind of place where I can ask for a short tour or just look at it, or will I be looked at like I'm crazy and it's the equivalent of asking for a tour of my local DMV? I don't know what to expect so these places have a bit of mystique to them and I'm very curious.
I do not know about US consulates, since US usually takes security extremely cautious thus the access to the consulates are strict.
However, I can tell from personal experience that consulates usually welcome their own citizens. I went to the Turkish Consulate in several countries, just to ask question (I was bored), and in all the times, they offered me beverages and more than happy to answer my questions. They tell me that I am welcome any day (Chicago Consulate). My friend goes there about once a month when he is in the area to enjoy some Turkish coffee and the amazing view.
In any case, you will need to have a valid ID (passport to be safe) to enter. And as I mentioned, US consulates are hard to get into and usually they work by appointment. There is no restriction however to ask the security as you mentioned it is close by.
From my perspective and in my experience (as a non-American citizen), American consulates are tightly guarded very unpleasant places where people often get mistreated.
As an American, you are less likely to be put through an awful experience, but I also find it very unlikely that it will be any fun. Note that much of the mistreatment I experienced was by local staff who abuse their position, not the Americans working there.
When you get there, you might expect very tall fences with sharp spikes and uniformed guards with machine guns. If it's in a crowded part of the city, there might not be room for tall fences, but the armed guards will be there.
At the consulate in Romania they won't let you pass in front of the building and will force you to cross the street. I am also aware of multiple occasions when they did not let people take any bags inside, no matter how small. They did not provide any place to safely store bags. Instead the (local) guard suggested that people should pay some of the taxi drivers idling nearby to keep the bags in the taxi's trunk. Now you can imagine that anyone who takes a mutliple-hour train ride to the capital city to apply for a visa at the consulate is going to have at least a small bag with their documents, wallet and phone in it; it's a necessity for such a trip. It's the sort of abuse of position I was referring to.
I am sure that American consulates are not quite as bad in all countries (though all of the ones I've seen from the outside looked menacing), but keep in mind that Romania is a safe country that is an ally of the USA and Romanians have typically pro-USA views.
Based on what I wrote above you can make your own judgement of how much fun it might be to visit one, even if you are an American citizen.
US Embassy Prague hosts a separate American Center with public events and cultural programming. Some other US embassies have similar programs. If your destination has one, you could pop in and/or attend a program if you want.
In general, I'd personally spend my vacation time exploring the foreign place I'm in rather than trying to visit my consulate.
(don't know if this can be considered an answer)
There is not much "fun" going on in consulates; and you just don't go there if you do not have immediate consular business.
Most of the employees do clerical work handling/managing liaisons for various agencies with the local country equivalent (mostly for cooperation on various levels, economic, scientific, cultural).
Most big consulate or embassies also have a "public" face for cultural events, either exhibitions or "parties" but they are usually held in different venues; for example, in Paris, the Canadian Embassy will host public events at the Canadian Cultural Center (same this for the US at the "American France Center".
In some instances, there are some embassies that are in historical building that will offer limited guided tours (for example, the French embassy in Rome at the Farnese Palace or the Canadian embassy in London).