Um, I grew up in East Boston and I have to give you a big NO on that one. However, Winthrop is nice. Winthrop is essentially East Boston With Trees, but it's definitely a big step up. Eastie.....it's not what it used to be and I wouldn't recommend it.
You can probably get a decent house in Winthrop with about $450,000 to spend. You just have to watch out for it because it's highly in debt and may be on the verge of receivorship which will shoot the property taxes through the roof.
I lived in Winthrop after Eastie and it was much better. Nice community. Close to the city. Close to transportation. Two thumbs up for Winthrop.