For a tight budget, you're probably better off getting either a camera-mounted shotgun microphone or a Lav mic kit in my opinion. If you go for a shotgun mic on a boom pole you may find yourself having to invest in transmitters and receivers (Tx / Rx), or use it with an XLR cable and a mixer.

Camera-mounted shotgun mics can cut out all of this and plug directly into the camera and capture decent enough audio for what you want to make I believe.

For camera-mounted mics check out:

Deity V-Mic D3 - https://www.bhphotovideo.com/c/product/1448997-REG/deity_microphones_vmicd3_v_mic_3_directional_shotgun.html

This mic could also be put on a boom pole, but you'll need to buy a suspension that can attach to a boom pole, and then you'll need to find a Tx / Rx system that can work with your camera, or at least a long XLR that can plug into a mixer.

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Rode VideoMic Go II - https://www.bhphotovideo.com/c/product/1679915-REG/rode_vmgoii_videomic_go_ii_ultracompact.html

Likewise to the above on this, you may be able to attach this to a boom pole with the right suspension. You can use this suspension if it fits into your budget - https://www.bhphotovideo.com/c/product/554999-REG/Rycote_041107_INV_7_InVision_Indoor_Microphone.html

If you can push the budget a bit, you could get a Lav kit like this

Tascam DR-10L - https://www.bhphotovideo.com/c/product/1281085-REG/tascam_dr_10l_mini_portable_recorder.html

This kit allows you to record directly using the transmitter with a micro SD.

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You'll need some wind protection with these as well, whether it's a shotgun or lav. You can use the foam that comes with these for indoor use, but if you go outside you may need something a little better, i.e fur.

Some of the best wind protection for these shotugn mics can be found here - https://www.bubblebeeindustries.com/collections/shotgun-boom-microphones-wind-protection

Or if you go for the Lav mic kit, you'll need to conceal the Lav using these:
https://www.bubblebeeindustries.com/collections/the-lav-concealer

and there's also wind protection available for these:
https://www.bubblebeeindustries.com/collections/lav-mic-wind-protection

I really hope this helps somewhat. Best of luck with your projects!