Disclaimer: Not all bikes are the same. This Instructable uses a road bike. You may need to follow slightly different instructions in order to remove the wheels from the bike.

Tools Required:

Screwdrivers / Bike Tyre Lever
Wrench
Replacement Inner tube
Air pump

Safety Info: Be careful while dealing moving parts, as you may get pinched

If you're confused at any step, check the photos provided. Captions have been added to some of them for further clarity.

Step 1: Removing the Wheel From the Bike

Flip the bike over so it is resting on the seat and handlebars.
If the tire is in the front, loosen the bolts on either side of the tire.

If the tire is in the back, you must remove the chain from the gear before moving on to the next step.
1. Using a wrench, unscrew the bolts on either side of the tire
2. Pull the chain off while rotating the pedals
Take wheel off bike frame

Step 2: Removing the Inner Tube

Take off the cap from the valve, and remove fastener
Stick lever or screwdriver in between the tire and wheel
Go around the wheel, taking the edge of the tire out of the rim. Note that you are not taking the tire off. Only one side of the tire needs to be taken out.
Pull out inner tube
Examine tire for anything that could puncture the new inner tube

Step 3: Inserting the New Inner Tube

Get new inner tube, make sure it is the correct size and type. You can find the correct size on the tire or the inner tube you just removed.
Insert the new inner tube by placing the valve inside the rim first
Insert the rest of the inner tube into the tire. Once the inner tube is half in, you can slightly inflate it to make this step easier.
Press tire back into wheel.
1. Tuck the tire back into the rim while turning the wheel. As you're doing this, be sure to check that the portion you've tucked back in is not coming out
2. When the tire is nearly all the way in, use screwdriver or prying tool to get the last bit back on. Be careful, as this is the most difficult step. If you push the tools in too far, you will risk popping the new inner tube.

Step 4: Inflating the New Inner Tube

Use a pump or air compressor to inflate new tire
Put the fastener and cap back on and you're finished

Step 5: Troubleshooting

There may be other issues causing your inner tube to turn flat. If your inner tube goes flat again after following this tutorial, it may have been faulty, or something on the tire / rim may be poking into the inner tube.