Learn Hardcore Japanese: watasi ha huyu ni ryokou wo simasu.

は is the topic marker. It tells us that 私 (“I”) is what we’re talking about, and everything that follows is a comment on that topic. In English we might translate it as “As for me, …”

Why is は pronounced “wa” instead of “ha”?

Although the hiragana is は, in its role as a particle it uses the historical reading wa. Whenever you see は marking a topic, say “wa,” not “ha.”

Why is に used after 冬, and what does it express?

When you attach に to a time word like 冬 (“winter”), you mark the point in time when an action takes place. So 冬に means “in winter.”

What is the function of を after 旅行?

を is the direct-object marker. In 旅行をします, 旅行 (“travel/trip”) is the thing being done, so we mark it with を to show it’s the object of the verb する (“to do”).

Why do we use 旅行をします (to travel) instead of simply 行きます (to go)?

行きます means “to go (somewhere).” If you say 旅行をします, you’re emphasizing the act of traveling itself (“to take a trip”). It’s like “do a trip” vs. “go.”

Can we omit 私は and just say 冬に旅行をします?

Yes. Japanese often drops the topic or subject when it’s clear from context. 冬に旅行をします still means “I will travel in winter,” and it sounds more natural if you already know who’s speaking.

Why does the verb します come at the end of the sentence?

Japanese generally follows a Subject–Time–Object–Verb (S-T-O-V) order. Placing the verb at the end is the standard pattern, though you can move time or object phrases around for emphasis.

What is the dictionary form of します, and how do you form this polite non-past tense?

The dictionary (plain) form is する (“to do”). To make it polite non-past you take the stem し and add ます, giving します, which covers both “do” and “will do.”

Sometimes I see 旅行します without を. Is that correct?

Yes. With many noun+する verbs (especially loanwords), you can drop を in everyday speech and writing:

旅行をします → more fully marked
旅行します → perfectly natural

Could I use へ instead of に after 冬—like 冬へ旅行します?

No. へ marks direction or destination (“to [a place]”). 冬 is a time word, not a location, so you must use に to indicate “in winter.”