If you are a permanent resident of the U.s. with a current, unexpired green card, you should be able to travel to Canada, England or the Bahamas - but you will also need a current, unexpired passport from your country of citizenship to do this. You will need to inquire at the Russian consulate about the procedures for renewing your passport.
Travel outside of the U.S. does need to be disclosed on a citizenship application. However, there are only a few ways that travel can be a problem for naturalization: at least half of the statutory period you have to be a permanent resident before applying for citizenship (five years, unless citizenship was based on marriage to a citizen in which case three years) must be spent physically in the U.S. Also, (and I'm simplifying a bit) your home must have been in the U.S. for the entire time, even if you visited abroad (individual, very long trips abroad could cause doubt as to where you were "living" vs "visiting"). Finally, where you visit has an impact - if a place is prohibited (Cuba, for instance, if no permit is obtained), or you traveled back to a home country and your green card was based on asylum - a claim you were afraid to go back there.
Of course, you also need to be sure to look into the rules of the place to which you want to travel to make sure you have the proper visa or other entry paperwork to go there.
The previous response posted to your question contains important information. In addition to that response, please note that if your travel plans may require you to be abroad for a long time, it may be worthwhile to examine whether you qualify for one of the exceptions to the general rules about absence from the U.S., such as travel necessitated for certain employees of the government, of some international organizations or of some American firms involved in international trade. Exceptions may apply to certain religious workers too.
[Note: Consistent with Avvo policy, this communication is intended as general information and not as specific legal advice, and this communication does not create an attorney-client relationship.]
David N. Soloway
Frazier, Soloway & Poorak, PC
1800 Century Place, Suite 100
Atlanta, Georgia 30345 www.fspklaw.com
404/320-7000 877/232-5352 email@example.com