You need very little strength training equipment to succeed in Full Range Strength and Flexibility Training. In the mid-1990s, after investing a good part of his life inventing and developing the Nautilus and MedX exercise machines, Arthur Jones wrote:
"Apart from a rather limited number of hardcore bodybuilders who are misguided enough to believe that they have a chance to compete against the outright genetic freaks that now dominate bodybuilding competition, just about anybody else in this country can produce nearly all of the potential benefits of proper exercise without spending much if anything in excess of about twenty dollars. You can build both a chinning bar and a pair of parallel dip bars for a total cost of only a few dollars, and those two exercises, chins and dips, if properly performed, will stimulate muscular growth in your upper body and arms that will eventually lead to muscular size and strength that is very close to your potential.
“Adding full squats, eventually leading up to one-legged full squats, and one-legged calf raises, will do much the same thing for your legs and hips. Using this very simple routine, when you get strong enough to perform about ten repetitions of one-armed chins with each arm, your arms will leave very little to be desired.
“Or, instead, you can do what many thousands of others are now doing and piss away thousands of dollars and years of largely wasted effort while producing far less results. The choice is yours.” 
As mentioned by Jones, to fully develop your potential, you need some strength training equipment. I agree with his basic list: a bar for chin ups and pull ups and a set of parallel bars for dips.
While as Jones said, you can go very far with nothing more than that, to that list I would add the following:
If you are going to purchase a free standing pull-up bar, you might as well get one that also serves as a squat or power rack. A good squat or power rack is an essential piece of strength training equipment if you want to do heavy weighted regular barbell squats to fully develop the strength potential of your hips and thighs. However, it is also useful for bodyweight strength training. A good squat rack will have a pull-up bar and a four-post rack will take attachments to enable you to do dips. A squat rack can also function as a stall bar. A squat rack can also be very useful if you want to do barbell overhead presses as a part of developing your ability to to handstand push ups.
I personally have a four-post Body Solid rack and add-on parallel bars (see below). However, I don't like the position and camber of the pull-up bar on this rack. If I were starting out, depending on my budget, I would either get one of the inexpensive two-post racks (Cap or Body Champ) and a separate dipping bars, or one of the two power racks (TDS or Titan) with parallel bar add-ons (see below)
Here are some examples of what is available on Amazon (Full Range Strength Amazon Affiliate Links). The two-post racks below all have been rated to support 300 pounds either as a barbell on the squat rack, or on the pull-up bar. That would be plenty for the typical trainee. The four-post racks with 2" square upright steel posts will accept add-on parallel bars.
You will need parallel bars for doing full range dips. If you get a two-post rack for pull-us and squats, you will need separate parallel bars. If you get a four-post rack with 2" square uprights, you can get add-on parallel bars. I have a four-post rack and use the TDS adjustable dual purpose handles for dips.
Here are some examples of parallel bars available on Amazon (Full Range Strength Amazon Affiliate Links).
In his book Building The Gymnastic Body, Coach Christopher Sommer stated that he believes that "Rings are the single greatest tool ever made for developing upper body strength" and "the only piece of equipment that is absolutely essential in your training." He points out that rings are strong, light, portable, incredibly versatile, and can be used anywhere you can find strong overhead support, indoors or out. That can be a high bar, chin-up bar, rafters, or tree branch.
Here are some examples of gymnastic rings available from vendors on
Amazon (Full Range Strength Amazon Affiliate Links). I personally use and really like
the wooden Bomba Gear rings. The wooden rings absorb sweat from your hands making it easier to maintain your grip without using chalk.
Paralletes are a virtually essential piece of strength training equipment for anyone who wants to develop impressive handstand skills.
Here are some examples of gymnastic rings available from vendors on Amazon (Full Range Strength Amazon Affiliate Links). I personally prefer the wooden bars.
A dipping belt gives you a safe, relatively comfortable way to add resistance, in the form of barbell plates, to your pull-ups and dips.
Here are some examples of strong quality dipping belts available from vendors on Amazon (Full Range Strength Amazon Affiliate Links).
A hyperextension bench enables you to perform back arch ups to develop your hips, hamstrings and lower back.
I personally use the Marcy Roman Chair/Hyperextension Bench. It is an excellent, very well-made, strong piece of strength training equipment. However, if you due to space limitations you would prefer a bench that you can fold up and move out of the way easily, you might like the Stamina Hyper Bench. (Full Range Strength Amazon Affiliate Links.)
Elastic resistance bands are very useful for performing exercises that would otherwise require expensive machines, such as leg curls. They are also useful for performing a variety of flexibility and mobility exercises. They can also be used to add resistance to dips, pull-ups, and single-leg squats. They are very portable and inexpensive strength training equipment.
I use the Bodylastics resistance bands in my training. (Full Range Strength Amazon Affiliate Links.)
Fractional plates include plates weighing 1/2 pound (or 1/4 kilo) or less. These enable you to gradually load exercises for smooth progress. This is especially useful for loading low-leverage exercises like hanging L- and V- raises.
For many exercises, it is impossible to successfully progress in jumps of 2.5 or 5 pounds. Making large jumps like this will result in stalling out. I find fractional plates to be essential strength training equipment.
I have a set of eight 0.5 pound plates and two 0.25 pound plates. I also have four washers each of which weighs about 0.125 pounds. This allows progression in increments as low as 0.25 pounds on standard 1" barbells and dumbbells, and as low as 0.125 pounds on exercises like L- and V- raises and the "windshield wiper" floor exercise. A set of 0.25, 0.5, 0.75, and 1.0 pound plates (8 plates total) with the same washers would allow the same.
Here are some of the options (Full Range Strength Amazon Affiliate Links).
A standard 1" barbell and dumbbell set totaling about 100 pounds will give you the small weights you can use for doing loaded stretching and for loading pull-ups, dips, and single-leg squat variations.
Many standard 1" barbell and dumbbell sets available these days have threaded ends with screw-on collars. I have a set of dumbbells like this. I don't like them; the collars come loose easily during use, creating a safety hazard; and they take longer to secure compared to the clips. The standard 1" spring clips don't work on this bars.
If you get a standard barbell and dumbbell I recommend getting one that has spring clips to secure the plates, such as the one below by Adler Sporting Goods sold on Amazon (Full Range Strength Amazon Affiliate Link).
An Olympic barbell set is essential strength training equipment if you want to perform standard full squats to develop the full strength potential of your hips and thighs. Most people will not need more than 300 pounds. If you get one of these, you will also want to have a tree to store the plates when not in use.
I purchased my first Olympic barbell set and plate storage tree used on Craigslist. Purchasing from Amazon will cost you a lot in shipping.
1. Jones A. My First Half-Century in the Iron Game, No. 68. http://www.arthurjonesexercise.com/First_Half/68.PDF