Heathrow to Bayswater

Heathrow CONNECT to Ealing Broadway, Tube to Queensway or Lancaster Gate depending on where your hotel is

Or Picc Line to Earls Court the District to Bayswater, then you are there not somewhere else.

Or if you have money, taxi from Heathrow or pre-booked transfer service
Heathrow Express (15 min) to Paddington and taxi

I would go with Heathrow Express plus taxi (or even walk if the hotel is near Paddington and you don't have heavy bags); or if you find that too expensive then as #1 or #2, but that will take longer and would not be so nice in the rush hour or very hot weather (as if...).

Why the HEX when the Heathrow Connect also goes to Paddington - and is far cheaper?

#5 It's a bit quicker, a very nice train with plenty of luggage space, and after an expensive long haul a few quid here or there is not such a big issue to me. You don't have to use if you don't want to, but sometimes it suits me - and obviously others too.
And sometimes I use a taxi/minicab/etc. and that costs far more. I guess you won't approve of that either :)

What do you mean by best? Cheapest, most comfortable or fastest?

#2's way= very cheap, but may be uncomfortable and slow
#1's way= less cheap, first train will be very comfortable and the second may not be
#3/4= very expensive, very comfortable and fast

Thanks all for your replies .Thay have been most helpful.

Quickest --> Heathrow Express, the most expensive train in the planet. But you would have to walk from Paddington or change to a tube for one stop or walk.

Otherwise, the tube, the Piccadilly line, changing to District and Circle.
(It feels like it takes for ever on the tube in, that is why I now prefer Gatwick.)