As others have said, there are a lot of different ways you could go. If you are unsure then you should start with a method that doesn't require any special equipment.
One thing you will find is that the type of brewing you can do depends a lot on how good your heating source is. If you have a good gas stove, then you can probably do a 6 gallon batch full boil. If your stove sucks then a partial boil might be better. If you have the budget to buy an awesome outdoor gas burner or electric brew kit, great, otherwise you just learn to adapt your brew process to suit the stove element that you do have.
Here are some things you can buy that will be useful no matter which way you decide to brew...
10 gallon stock pot or 6 gallon if on a budget
Starsan
Hydrometer
good digital thermometer
digital scales
bottling bucket or second fermenter with tap
racking tube or autosiphon for transferring beer between vessels
All of those things are reasonably cheap, and will be useful for any brewing type.
If you want to go all grain, you should also start with...
grain bag to fit your kettle
Copper immersion chiller - these are a bit expensive but make brew day easier and faster.
Most types of extract brewing don't need the copper immersion chiller so this is one reason why extract is good to start with before moving to all-grain.