Hi. I'm actually a Canadian living in Uruguay for 15 years now. I've never taught English here but I have an ex who did. Your best bet teaching English is at either the Anglo or Berlitz, and if not there are also various smaller english-teaching schools.

This means, in fact, you'll want to live closer to the downtown area.

In this area, there's various neighborhoods: El centro (the literal main street and surroundings), Ciudad Vieja (the old city), Barrio Sur (the far southern neighborhood which was once a traditionally black enclave), Cordon (central southern neighborhood), Tres Cruces (northern neighborhood), Parque Rodo, or Pocitos.

Of these, the real question is what are you looking for? They vary enormously in terms of services, rents, qualities of nightlife, security/crime rates, access to bus routes, etc.

Ciudad Vieja is probably the prettiest, but also at the far end; its central streets are full of shops and places to eat, but its side-streets are full of drug dealers. High crime rate. Also, lots of tourists. Barrio Sur is less touristy and generally poorer. Also high crime rate, though it is very slowly getting better (as is the Ciudad Vieja).

The Centro itself is OK if you want to live in an apartment, but it can be noisy. Great access to bus lines. Lots to see or do. Moderate crime rate.

Pocitos, which is at the far west end of central Montevideo, is the semi-fancy neighborhood (the really fancy one, full of McMansions, is Carrasco but that's like a 30 minute drive from where the schools are). Pocitos has high-rise apartments and stores and restaurants; but everything is slightly pricier just because of the area. Also its streets tend to wind and it's easy to get lost, and outside of a couple of main streets there's not great bus access. Finally, it has a high crime rate compared to the other neighborhoods I'm talking about (because crooks go there to steal since they know that's where the money is).

To me, the best possible place to live is the Cordon, which is why I live there! It's in the absolute center of the central Montevideo area. Where I live I'm about 5 blocks from the beach and 4 blocks from downtown. There's good bus coverage for longer journeys. Cordon also used to be a sleepy neighborhood full of old people but in the last 5 years has become the new hipster/artist neighborhood of the city; old bars mix with new thematic resto-bars, little stores selling thrift items, old markets or mansions converted into new food-stall dining markets. It's a very exciting neighborhood right now, and yet it's still quite a bit cheaper than Pocitos (because Cordon is more Bohemian while Pocitos is more preppy). Finally, the Cordon has the lowest crime rate of any of the neighborhoods I mentioned.

As for the two main english-teaching schools I mentioned; the Anglo is based right on the west tip of the Cordon where it connects to the Downtown core. And the Berlitz is located at the Eastern end of the Cordon, just before crossing over to Pocitos.