If you know Russian, use it. Despite the political situation, Georgians are smart and open minded. They may not like the Russian government but they love Russian people and they love the Russian language.
I'm pretty convinced after seven months in Georgia that people learn Russian there for just the same reason foreigners everywhere learn English: it's cool and it's useful.
English is spoken, but I wouldn't say "widely". It's spoken by people aged in their 20s and younger, moreso if they're in a big city and moreso depending on their level of education.
In 2010 the Georgian government made a big push for the country to learn English and has been recruiting thousands of native English speakers, mostly from USA, with the goal of having at least one English teacher in every village.
The result is that young kids speak Georgian, a few English words and phrases they've learned at school, and a few Russian words and phrases they've picked up from friends and relatives. People in their teens and twenties are pretty fluent in both Russian and English with just a Georgian accent in both. People in their forties or older rarely speak any English at all but will speak fluent Russian with an accent. (People in their thirties seem to be a bit of a grey zone between the 20s and 40s age groups).
There are also a considerable number of ethnic Russians of various generations living especially in the cities in Georgia. Plenty of ethnic Armenians and Azeris as well as expats live in Georgia too and communicate between each other in Russian.
- If you know some Russian, use it first. It will actually endear you to Georgians more than English at this point. Taxi drivers know Russian but no English. Restaurants aimed at rich tourists have some English speaking staff but in small authentic restaurants and dukanis only Russian (or Georgian) will be useful.
- Use English when your Russian isn't good enough if the other person knows English.
- Georgians are very very impressed with any foreign travellers who can speak a few words of Georgian. Especially in places few tourists go they will be particularly amazed and delighted.
You can get the basics from searching the Internet for videos. Just knowing the numbers one to ten, "hello", and "thank you" is a reasonable goal. It's much more important to learn the Georgian alphabet, which is nothing like the English or Cyrillic alphabets, and in fact not like any other alphabet. The last time I looked not many of these Internet tutorials went much beyond the basics anyway. You can buy imperfect but usable phrasebooks in Tbilisi.
Some more tips:
Being fluent in Russian won't prevent you from getting ripped off in the bazari but it will make it a bit easier to haggle. Anywhere except the bazaar prices are marked and fair and where they're not it's easy to haggle by smiling and holding up fingers (-:
If you visit the Armenian-majority cities, don't try to use Georgian. Stick to Russian or win hearts with a couple of Armenian words.