I can suggest several thing:
Fuel filter needs replacing
Dirty air-intake filter
dirty fuel injectors -
spark plug wires not connected properly
that's all I have!

In addition to Klinger's suggestions, let me give you a different
perspective, based on my recent experience.

I also have a 2000 Malibu V6, which recently started some weird behavior.
The fuel gage and temp indicator wouldn't work after starting the car, as
well as the heater fan, electric windows, rear defroster. After 5-10
minutes, some of them would work again - sometimes all of them. After
compiling a list, I could see that all of the items affected were
accessories that only worked with the ignition key ON. After asking a
neighbor who works at a Honda-GM dealer (he's a Honda guy), he asked a few
of the GM guys if it could be the ignition switch. They all said yes. So I
replaced the switch (list $140, discount $100) and it cured ALL of the
problems.

OK - how does this link to your problem? According to the wiring diagram,
all of the engine operation systems - injectors, ignition, emissions, etc -
are also ignition key ON items. The ignition switch has five major sets of
contacts in it - in my case three of them were damaged/corroded from current
arcing to the point where they restricted current flow. It's possible that
the contact(s) in your switch that energize the ignition/fuel systems are
also damaged.

Hope this helps.

doug

I had that happen to me on another GM car and it turned out that the
ignition was shorting out through a wire that had rubbed against the
steering column.  Every time I would go over a bump the wire would
shift and short out.  After a few minutes, the car would start and
everything would be OK until the next bump.

Just a thought.