Plunging won't take care of everything. After we bought our house many years ago I used a wire coat hanger and found a rose stem lodged in the toilet drain a few inches beyond where I could reach that was catching 'things'. Usually it flushed ok but sometimes didn't, probably when a lot of paper or whatever had caught on the stem. The vent is a possibility but if it flushes normally sometimes probably not. I would think the vent would either always be plugged or not. You could try snaking it though you have to be careful not to damage the toilet. Last thing would be to turn the water off, flush, sop up the remaining water in the tank, take it off, take the bowl off and check the trap. Put in a new wax ring and put the toilet back on. I'm sure there are plenty of videos on the internet on how to do that.