What is the best kind of electric guitar and amp should a beginner buy?

Best bang for the buck you can get is an Ibanez guitar and amp package. Another good one is Fender Squire series.

If you decide to stay with it, then you can spend more money and upgrade to mid or upper line guitars and amps.

You can get either of these at Guitar Center or Sam Ash and not spend a lot.

I would get a “class A” amp, which typically has one or more tubes. The small ones (see Crate amps) are not that expensive. They have a great sound. When you get good, you can continue to use this amp in the studio, or mic it for live performances.

Otherwise, you’d be buying something that sounds like junk and you’d have to get rid of it sooner or later.

One thing that beginners overlook is this: WHEN IT COMES TO ELECTIC GUITARS – THE AMP IS CONSIDERED PART OF THE INSTRUMENT. A great amp can make any piece of junk guitar sound amazing. A poor amp will cripple even the finest guitars. YES – THE AMP REALLY MATTERS – A LOT!!!!

I would recommend a Fender Squier Set or First Act.
If you have a Guitar Center Store around, sometimes thay have sales on the Fender. It comes with an amp, guitar- red, black, or blue, a strap, cables, picks, book,and extra strings- all for around $250. First Acts are cheap, but not sure how good they are.
I have a Fender Squier and it is awesome- great sound, and it stays in tune for awhile, but I would recommend buying some type of pedal with it.