I know things... A lot of things.
Join Date: Feb 2015
Motorcycle: GSXR 750 K4
Re: Please help, complications starting engine
Check the idle speed once the bike is warmed up. If it's lower than what is specified in the service manual (1200 +/- 100 rpm) adjust accordingly. Also note, after getting the bike to start, whether it idles higher than that (1500-2000 rpm) for some time (15s to more than a minute, depending on a number of factors, most notably ambient temperature). If not, it should be investigated. The battery should also be checked: search the site for "measure battery voltage DMM". Try looking up other similar models on YouTube and listen to the cranking sound, trying to spot if yours is noticeably lower in pitch, implying a slower-turning starter. If all these check out, the condition of the spark plugs should be inspected, or, since the bike is new and you have only the previous owner's word as to its state of maintenance, they might as well be replaced preemptively.
Finally, after starting and idling properly, does the bike otherwise perform normally? This question assumes you have enough experience with this, or similar bikes to be able to determine "normal" operation with some confidence. I'm asking because, for instance, if the engine runs fine at small throttle openings (say going at low speeds, pulling off from intersections, etc.) then problems with say, the vacuum lines can likely be ruled out, otherwise they might need to be investigated.