Estonian foreign policy is inevitably a David & Goliath affair. From our position in the EU we do work with small countries that - well, want to be us, basically. But most of the really important things our diplomats do are relations with our allies, usually the US and the big EU countries, and the enemy-apparent, Russia.
Giustino mentions an idea which I think should be embraced - an Estonian version of the British Council or Goethe Institute or such; tasked not so much with promoting language study or cultural exports, but with generating public goodwill towards our country. There is already a tradition of Estonian Houses as community centres. If Hemingway was right, and indeed in every port in the world you will find an Estonian, the resource is ready to be called into public service.
Estonia would very much like to be like Switzerland - just friendly enough with its neighbors to not be involved in any of their conflicts, otherwise left alone; "we'll take your money, but please don't move here". Of course objectively this is impossible - Estonia will always and inevitably be aligned with some major force, and all it gets to do is choose the one it fancies more. Exporting counterpropaganda and actively developing goodwill around the world is an important task.
However, and in conjunction with the Russia crisis, I think people tend to underestimate the Estonian government's ability to manipulate others. The way to make Europe care about Estonia's security is to make Europe worry about Russia's aggression. Take the border treaty debacle: with the benefit of hindsight, we might come to the conclusion that it was a great diplomatic victory for Estonia, provoking Russia into a hysterical, unreasonable response. And it's the same with the Bronze Soldier mess: if we give in to the conspiracy theory, logical analysis suggests that Estonia pulled off a major coup, forcing Russia into a move necessitated by internal issues, which in turn left the EU with no choice but to call the Kremlin on it.
Estonian politics are small country politics; small countries cannot afford ideals. They have goals and purposes, and their primary purpose is preserving the nation. Ideals are something you sacrifice a lot of human lives to defend. In Estonian foreign policy, the ultimate goal of ensuring the security and prosperity of the country must be pursued by the sneakiest of means. The fact that Estonia is not a significant military power or economic contributor to the EU budget does not mean that Estonia's diplomats and policy chiefs are not successful. They just go about things in a slightly different way.
Russia is in the habit of making threats from a position of weakness; we must convince the EU and NATO that Russia must never be allowed to come into a position of strength.