Why is בְּרֵאשִׁית used in Bereshit 1:1 but תְּחִלָה used in Shoftim 1:1? Both words mean beginning/first as far as I can tell. I am just wondering why one is used in one location and the other one in the other location.
The Malbim explains that תחילה applies to time, negates previous conditions and is regardless of following events. ראשון applies to sequence (ie. there will be following events) and doesn't imply a negation of anything previous to it.
He explains that תחילה wouldn't be appropriate to use in place of בראשית because it negates any time before it. There was no time before creation to negate, so תחילה would be inappropriate.
To my knowledge, תחלה refers to a general start to something, whereas ראשית means strictly the beginning (because it is from ראש and ראשון).
I do not have a source for this, but my father always told me that it is interesting that בראשית can also mean, with a few changes, "in the head of." This might come to teach us that G-d methodically created the world, by planning it all out before making it.
Hope this helps.
As the OP implies, "b'techilas" and "b'reishis" have similar meanings and might be used to properly describe the same general idea without either being a wrong usage.
That being said, lets narrow down the technical difference between them.
"techila" means "start"
"rishon" means "first" or "head"
"techila" in the case of Shoftim 1:1 simply is asking which tribe should start the invasion. Replacing it with "rishon" would imply that the tribe who starts is part of a chronological progression in which the next tribe to battle is called "second" and all the other tribes wait until they fight as well, then the "third" tribe gets to attack etc. It would describe a parade.
However, Shoftim 1:1 simply has Judah attack first. The rest of the tribes then proceed in no particular chronological parade order into the land. They may or may not have been simultaneous. So there was no order of tribes as a group divided into place units. Therefore, a simple "techila" is a better choice of wording here.
In Genesis 1:1, the technical translation (see Rashi) is "In the beginning (1st stage) of ("Bereishis") G-d's creating of the heavens and the earth.."
Replacing this with "At the start of ("B'techilas")G-d's creating of the heavens and the earth..." would technically fit. However, G-d is teaching that his creation plan is a set of orderly days in which each day is assigned its chronologically dependent content. Therefore the word that means "first" ("In the first stages of G-d's creating of the heavens and the earth..") is a better choice of wording here. You will notice that this is true, since each day is called "first second third...etc.
Rashi (due to grammatical concerns (preposition "of" not fitting well)) offers a second pshat (explanation) based on a Medrashic hint.
"B'reishis" -- "For (the sake of) the "Beginning", (that's why) G-d created the heavens and the earth."
Now the sentence reads better gramatically. The Medrash points out verses in Proverbs and Jeremiah which refer to the Torah and Jews as "reishis" or "firsts". The Torah is the first of G-d's path and the Jews are the first of G-d's crop. Here the Medrash views the meaning of "reishis" as "head" or "chief".
"For the sake of the Torah and the Jews "B'reishis" G-d created the heavens and the earth."
If G-d did not use "reishis" instead of "techilas", then the verse would not be a link to the other two verses that also use "reishis", when describing the Torah and Israel.