In English, verbs aren't commonly described as infinitives except by people who are learning English as native speakers of another language (especially Romance languages) or by people who are obsessed with Latin. When people do talk about English infinitives, they're talking about verbs that take their simple, present tense forms and do not have a definite duration. For example:
- I want to go to Florida.
- To resist is futile.
- I could read all day!
The infinitive structure comes to us courtesy of Latin, and its useful to English language learners because many Romance languages employe the structure. For example, verbs in Spanish have actual infinitive forms that get used in specific circumstances in Spanish grammar, like when you talk about something that pleases you. When I say in Spanish, "Me gusta jugar a baloncesto," what I'm literally saying in English is, "To play basketball pleases me." This sounds odd in English because we don't use infinitives the same way Spanish speakers do, but in Spanish, this kind of sentence construction is common.
In your examples, all of your sentences are using "travel" as an infinitive, but trying to figure out why will probably confuse you for no good reason. Learning how to analyze sentences can help you identify infinitives, but learning how to identify infinitives will almost certainly not benefit you in any meaningful way simply because we don't use infinitive structures consistently in English.
Here are the answers to the questions in your title:
- The infinitive form of a verb, in English, is the form of the verb used to describe action of a non-finite duration, possibly with the word "to" attached to it, as with "to go", "to play", or "to study".
- The bare infinitive form of a verb is the same as the infinitive form, only without the word "to".
- The base form of a verb is incidentally the same as the infinitive--the form of the verb used to describe action of a non-finite duration, without "to". We use the phrase base form to describe something different, though. A verb's base form is the form of the verb that has no special endings. You'll find verbs listed in the dictionary by their base forms. "Go" is a base form, but "went" is not. "Play" is a base form, but "to play" is not. "Study" is a base form, but "studies" is not. When you see a base form of a verb in writing, it's probably either acting as a first or second person singular verb ("I go," "You go,"), a plural present tense verb ("We go," "They go,"), and/or an infinitive. Base forms of verbs can be infinitives, but they aren't always.
