Learn Hardcore Japanese: watasi ha tomodati to ryokou ni tuite hanasimasu.

について is a compound particle meaning “about” or “concerning.” When you want to talk about a topic (in this case, 旅行, “travel”), you attach について directly to the noun:
• 旅行について = “about travel.”
It indicates the subject matter of your talk.

Why do we have 旅行に plus ついて? Isn’t that two particles?

In modern Japanese, について is treated as a single unit (“about”). Historically, it’s made of the particle に plus the continuative form of the verb つく, but you don’t split it in writing or speech. Always write and think of it as について, not 旅行について separately.

Why is 旅行 followed by について instead of the direct-object marker を or the location marker で?

The verb 話す (“to speak/talk”) doesn’t take its topic as a direct object. You don’t say 旅行を話す. Instead:
• Use について to show the topic you’re talking about (“talk about X”).
• で would mark a location or context (“at travel” or “by travel”), which doesn’t fit here.

What role does と play in 友達と? Could we say 友達に or 友達で?

と after 友達 means “with (someone).” So 友達と = “together with a friend(s).”
• 友達に would mean “to a friend” (indirect object) – not correct here.
• 友達で would indicate “as a friend” or location/context – also not what you want.
You can even expand it to 友達と一緒に if you want to emphasize “together.”

Why is 私 marked with は? Can I omit 私は altogether?

は is the topic marker, so 私は = “as for me.” It sets 私 as the topic of the sentence. In Japanese, if the speaker/subject is obvious, you can drop 私は, giving simply:
友達と旅行について話します。
That still means “I will talk with my friend about travel.”

What kind of form is 話します? How would you say it in casual speech?

話します is the polite (ます-form) present/future tense. It corresponds to “will talk” or “talks.” To make it casual/plain, switch to the dictionary form 話す:
• Polite: 話します
• Casual/plain: 話す

Is Japanese word order fixed? Could I rearrange 友達と旅行について and 私は?

Japanese is relatively flexible because particles mark each element’s role. You could say:
• 私は友達と旅行について話します。 (standard)
• 友達と旅行について私は話します。 (topic shifted)
• 旅行について友達と話します。 (focus on what you talk about)
As long as you keep the particles (は, と, について) with their nouns, the meaning stays clear.