I think that "die Reise" is a bit broader in meaning. "Die Fahrt" seems to be more specifically a road trip by car, bus or train, but maybe it has become broader in meaning with time. Someone correct me if I am wrong.
"euer" or "eure" is the informal plural possessive adjective (German still calls it a possessive pronoun, but says it is in attributive form which means it describes a noun.) for the Nominative case "ihr". At the beginning of a sentence. this would be capitalized. The attributive form changes to match the case, gender and number of the noun that it is describing.
"Sie" is the current formal form of you in Nominative case which becomes "Ihr" in Possessive form.
Yes, this is confusing. See this page for the Nominative case with possessive forms: http://www.canoo.net/services/OnlineGrammar/InflectionRules/FRegeln-P/Pron-Poss3.html
Here is a page which describes the differences between the various Nominative forms and once you understand that then go back to the possessives and memorize them. http://german.about.com/od/grammar/a/Germanyou.htm
Then again, you are right that "ihr" was used in Capital form "Ihr" as a formal form before "Sie"
http://german.stackexchange.com/questions/12402/ihr-as-second-person-singular Apparently the honorific form "Ihr", whose possessive is "Euer" or "Eure" etc., was replaced with "Sie", whose possessive is "Ihr" or "Ihre" etc., in the 19th century. You could encounter this form in books, but it would sound archaic in speech.
Euer takes different endings depending on the case and gender of the noun it's used with. In this case it's the akkusative case and the gender of the noun (die Reise) is feminine. The ending for this is -e, so that would be euere but when it's like this the second 'e' is dropped, so you have eure. Check out the wiki page for possessive pronouns and declension.
euer is the second person plural possessive and euch is the second person plural pronoun for the accusative and dative cases. Ich (nom) habe euch (dat) etwas (acc) mitgebracht - I (nom) have brought you (plural, dat) something(acc). (...) represents the respective case. Ich habe ihm eure Katze gegeben. I have given him your cat. Does that help at all?
The capitalized formal form "Eure" is archaic now as "Ihr" has been replaced by "Sie" which would use "Ihre" here for formal form. The form used here is not capitalized "eure" and would only be used for informal plural possessive form of "your" while the informal singular of "your" would be "deine" for "du". Scroll up for some links that I gave to someone else above.