Those are called the dakuten and handakuten. You can consider these modifiers to the base syllable. Generally these marks change where the sound is generated in your mouth (mostly). So か (ka) is more or less in the middle of your mouth, が (ga) is from your throat. Another explanation is you can think of them as modifiers to the “softness” and “hardness” of the syllable. So ひ (hi), か (ka) would be standard, び (bi), が (ga) would be softer (rounded), and ぴ (pi) or ぱ (\pa) would be harder (sharp).
For the syllable combinations, I don’t recommend that you study them individually, just get used to the initial set first. You’ll notice that they just also act as modifiers to the main syllable. So if you learned き (ki) and や (ya) separately, when you see きゃ (kya) you combine the pronunciation of the two syllables.
You might also still be coming from an English facing perspective and trying to assign syllables to the English counterpart in Roman script. So you assign か to ka (literally, k and a) instead of the sound か itself (i.e. to the letters ka specifically, not the sound), but it might be better to assign the sound to the character instead. This might be causing you a lot of confusion or trouble with memorization, because of this I would also recommend watching pronunciation videos while you memorize your hiragana and katakana. Memorizing the sound is the most important part to making sure you don’t bring your native speaking accent into the pronunciation.
If you know IPA you can also check out this Phonology Chart.
Thanks to Japanese being a syllabic language, there are no variations in pronouncing the individual characters. Unlike English with a (pronounced eh, cable) or a (pronounced ah, apple) (apologies to the folks that know IPA out there for butchering this), there are no ifs and buts when something is written out in hiragana or katakana, so what you see is what you get (barring misreading issues). So か will always be か wherever and whenever you read it.
This is unfortunately the hardest part of Japanese, simply because it will take years to reach this point. This is because you will have to memorize a lot vocabulary and kanji. The reason you’re able to read without needing to focus with the language you grew up with is because you grew up with it. Imagine how much time you spent practicing and reading in your native language before you got to this point. Fortunately, we’ve now got tools like Anki, WaniKani, Bunpro, etc. to supercharge how fast we get to that point. But overall it’s just going to take a bunch of effort and time because of the sheer amount of things to remember.
I like to think of something like flight time for pilots, where time in the cockpit while the plane is in the air is the primary measure of experience. In Japanese this would be time immersing in the language with videos, reading, listening, speaking, etc. So the more time you spend “in the air” the more instant that word “house (家 or いえ)” is going to be for you.