: Can my fingering be improved? I'm a beginner at piano, was wondering if my fingering for this piece can be improved Can my fingerings for the chords be improved? The melody part? I'm especially unsure about the section

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For the right hand: Everything looks ideal except for the red numbers. I would avoid the 5-4-3 since it feels unnatural to me, especially the 4-3 part. I would use one of the alternative options provided in Aaron's answer.
For the left hand: Consider 421 for the F chord at bar 6. For me it makes more natural the transitions to and from the F chord.

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So from the looks of it, the fingering seems fine. You could use 4,3 for the fingering for the RH in the second measure. But in general for most of the songs you are playing now, just try to use fingerings that minimize how much your hands moves as much as possible. This will help as you start to play more advanced pieces with faster runs.

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I agree that your fingering looks ideal for both left and right hands.
In light of your question about the "red-fingered" notes, here is some discussion of considerations that go into choosing a fingering.
Since you specifically mentioned a thumb-turn, here's a proposed fingering.
X:0
T:Happy Birthday excerpt
M:none
K:C
L:1/4
G/2G/2 g e c B A
s: 1 1 5 3 1 2(3) 1(2)

The parentheses indicate alternative fingers.
As Haversine said, ideally one minimizes hand movement; the more efficiently you can play, the better. However, comfort for your hand is a major consideration as is the intuitiveness of a fingering. The relationship to phrasing is also critical.
So some discussion of why you might choose or reject fingering options in the "Happy Birthday" "red fingers".
Key considerations
(pun not unintended)
The important moments affecting the fingering are the E and the B: the E because it's where you decide on consecutive fingers versus thumb-turn, and B because in the thumb-turn option, it's the best place for it -- it allows the thumb-turn to be coordinated with the arrival of the left-hand chord, which I find the most intuitive placement for it.
Fingering given in OP
This is quite nice, because you just use each finger in order, so easy to conceptualize and reasonably easy to play. "Reasonably" easy, because some might find the span from 5 to 4 to be uncomfortable or unintuitive.
5-3-1-2-1 (comfortable, but thumb-turn)
This option allows 5-3-1 to play the descending C major chord (G E C), which both fits most hands well, plus it's the "standard" fingering for a major chord. On the other hand, a thumb-turn is required. Good for a player comfortable with it; less so for someone not there yet.
5-3-1-3-2 (comfortable, but phrasing consideration)
For me, personally, 1-3 is more comfortable here than 1-2. So for someone like me, this has a nice feel to it. There's an implication for phrasing, though: unless you can comfortably span a sixth with 2 and 5, this fingering more or less requires a break before the final "Happy birthday to you." It's common to put a break/breath there, so this fingering facilitates that phrasing. However, it's a disadvantage if you want to move fluidly without a break/breath.
5-3-1-3-1 (compromise)
This would be a compromise between the previous two options. The 1-3 is for someone like me, who prefers it to 1-2, and the 3-1 facilitates the leap to F.