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Most used tool laying wood flooring will be a chop saw,
a track saw and a good jigsaw are also pretty useful/indispensable (depending on your level of frustration at using the wrong tool for the job)
I’ve never felt the need for a table saw.If you’re taking the skirting off (I recommend) to hide the edges then you could do the whole job with a jig saw and a hard-point.
cutting the mouldings not the flooring (multi tool or hard-point and a scrap of flooring as a spacer and some skill) will give you a much cleaner result going around doors etc.
I was thinking a table saw would be the best single main tool, plus a jigsaw or multitool for cutting weird shapes. If buying several then a mitre saw for cross cuts plus table saw for rip cuts would probably be best, but trying to be as budget conscious as possible (hence doing it myself rather than must paying a professional).