how to increase the speed of a wifi router?

This is like asking how to increase the speed of a Ford.  You 'could' spend $100,000 or more for a whole new engine and drive train.  Or just buy a new car for $20,000.

You buy a new router, silly!  There is no slow, medium, or fast setting on the outside of your router!!!  it is designed to go as fast as possible right out of the box.  After three years - you increase speed by taking a nice new one out of the box and enjoy the faster technology.

That simple.

Buy good speed internet

No idea, what is your question?

Use a new password and reboot regularly. Get a signal booster.

The only way to speed up internet is to buy higher speed internet.

First you have to find out the maximum speed of your router, which will depend on its transmission frequency.  There is a limit to the speed of any traffic over radio waves (which is all that WiFi really is) based on the "carrier" frequency.  You will NEVER EVER get more than the limit for that frequency.

As part of that, remember that your transmissions involve a protocol that includes an "end to end" component.  So what REALLY matters isn't always the WiFi hop.  If you have a gigabyte-capable WiFi but a 100 megabit cable TO the router, your speed is never more than 100 megabits.

Second, you have to determine your network load.  While there is no hard and fast rule about this, I have found after 28+ years with the U.S. Navy as a system admin and network admin, if you are using any typical network, if it is exhibiting more than about 30% utilization, the probability of network "collisions" goes WAY up and you start having to retransmit a high percent of your messages.

Third, what kind of speed does the OTHER end of that WiFi "point-to-point" connection have?  It, too, is part of the end-to-end nature of the connection.

Fourth, and this is particularly true in neighborhoods with many folks having their own WiFi hot-spots, how many other folks are banging away on their own routers?  That, too, is part of "network load" but more commonly a problem to home WiFi than to business WiFi.  If you can find a WiFi channel that is used less than the others, you might - stress MIGHT - see some improvement by switching channels.

You can speed up your wi-fi router by - positioning your router in the perfect spot, or by keeping it away from electronic devices, or keeping it separate from wireless signals, or by using a password, or by rebooting your router regularly.

Give us more information as to why you ask the question.  

As others said, you can't increase the forwarding performance of the router, but WiFi is subject to all sorts of gremlins that can make it work slower than it could.

generally, you buy a new and better one

You cannot improve the hardware and signal broadcast of a wireless router.  What comes from the factory is it.  You might investigate keeping its firmware up to date by refreshing the BIOS (if possible).  But that is only for security, perhaps additional features like VPN subscription capability.  No better radio signal performance.

You can do much outside of the router itself.  Try from this list to see what works for you (cheaper ones first).

--Depends on the location of that "wifi router" (elevated is better), what is between it and the signal destination (steel reinforced walls, large metal appliance cabinets, coated glass windows block radio waves), and the range (closer gives stronger signals).  

--You can download a wifi signal analyzer to a smart phone or a laptop.  Walk around and see the signal strength.

--If another wifi signal is overlapping yours, move the channel of yours to a quieter one.

--Beam the router antenna signal to another room with a radiation reflector.  A bent metal pie pan or unfolded Pringles can would reflect the radio waves into a stronger but narrower path.  Must be accurately aimed.  See Google how to.

--You can extend the range by using a PowerLine or HomePlug pair of relay devices.  They use the house AC power wires to send the signal to a different area.  One end device has an Ethernet cable from the router.  The other has a wireless antenna, or both can be with Ethernet.  Modern ones are quite fast.

--More trouble but better is to run an Ethernet cable through the attic or walls to a wireless access point (a network box with Ethernet in, antenna out).  Even another wireless router with its DNS service disabled would be fine for this.

--Get a wireless router that has both 2.4 and 5 MHz frequencies.  Then if your laptop has capability of both, you might get greater speeds.

I have tried all of the above, found the last two to be quite good.