Your description of what's going on doesn't seem consistent.
For a toilet to bubble while taking a shower implies there's a main line partial obstruction. If it were total, the shower would back up. WHen you flush the toilet, it tries to dump a lot of water fast, then stops. If it goes down, as you said, the blockage is not complete. But, you still have a blockage somewhere. Since it does drain, there is at least some room in the pipes for some water to be added when you flush. If, after not draining anything or flushing, you then try to flush the toilet, if the clog is a ways away, the toilet may flush, but a second or third try within a short time may fail since what was there has not had a chance to drain away yet. How many flushes could give you an idea about how far away the clog is.
If liquids, but not solids will flush in the toilet, then that implies something caught in the trapway that is allowing the liquids to pass by, but when there are paper and solids, they get caught up on the obstruction. Common ones are a comb, pen, pencil, crayon, small toy, etc. Those probably will not be removed with a plunger...you may need either a toilet auger, or to pull the toilet and then try to remove it.
SO, sort of sounds like you have more than one problem.