Your first table-tennis racket
In the golden age of American table-tennis (1940s thru 50s'), USA won the men's team event at the World Championships and also placed US men & women in semi-finals in the singles events. During that time the rackets did not have any sponge and the racket just had a sheet of rubber with the pimples (pips) pointing outward (pimples (or pips) made contact with the ball). However after the introduction of sponge in 1952 in Bombay (now Mumbai) World Championships by the Japanese, the sport of table-tennis has drastically changed irreversibly forever. Unless you are age 60 or over the use of old pimpled out rackets (known as hardbat) is strongly discouraged if you want to be a competitive player in tournaments ( the pimpled out rackets are still legal but now serve some special styles only......also keep in mind that sandpaper rackets are illegal in regular events of USATT / ITTF tournament ) .
That said you are strongly discouraged from using a pimpled out racket (even with sponge underneath) if you are just starting out. This applies even more to children. You must always start with a racket that has that has pimples inwards...meaning the surface that contacts the ball is smooth.
The strokes executed with pimpled out rubbers (especially without sponge) are very different from those executed with smooth (inverted) rubbers. In modern table-tennis (unless you later find out that you are more suited to a pimpled rubber style) , everyone especially children must start with or switch to smooth rubber.
Many players in American basements get so used to playing with pimpled (pips) out rubbers with no sponge and get a rude awakening when they play at a club. The chances that you may do well in a club with a pimpled out rubber racket are very minimal (if you are just trying to move up from the basement "ping-pong"). Many players refuse to accept this and when they first switch to smooth rubber, they lose badly. If you are talented you would adjust in a few weeks or months but definitely not in a day because the strokes are so different. Most basement kings (& queens) do not understand this and go back to their basements and stay happy as long as they can beat their mother-in-law (or father-in-law)
Such rackets in the world of professional table-tennis are known as inverted rubber covered rackets or smooth rubber covering rackets. All new players must only start with inverted rubber covered rackets if you are serious about competing. There is still a good chance you may switch to a pimpled surface one side only or both but this comes well after you discover your playing style.
Also tournament rules require that one side of the racket be red and other side black. (Both sides cannot be both red or both black) . All other colors such as blue , yellow , green etc are illegal in regular events of USATT / ITTF tournament play.
So if you want to become a serious player, do not purchase cheap fancy colored rackets or pimpled out rackets. Also cheap $10 or $20 rackets are useless
You can buy decent rackets two ways :-
1. At an established local sports store such as Sports Authority or Dicks etc . Limit yourselves only to brand names Stiga or Butterfly or Yasaka in these stores. Not others. Also do not buy the lowest quality of these rackets. Buy medium or higher quality rackets only.
2. You can buy via mail-order or eBay or Amazon etc. While a racket is super extremely important in tabletennis , no need however to buy a $300 racket to start with. It will not help you much, if any. Budget may be $50 or less. Goggle / Yahoo Search for table-tennis racket stores on internet and order from them . When you mail order you can buy additional Chinese brands such as Friendship , Double Happiness, Double Fish etc and brands such as Tibhar, Armstrong, Killerspin which will be of high quality in addition the store brand names mentioned in item 1 such as Butterfly, Stiga & Yasaka
Keep in mind these key factors below
1. Buy only smooth (inverted) rubber covering rackets
2. Do not buy any pimpled out rackets (with or without sponge). I play with a specialized pimpled out rubber on one side of my racket, but in my opinion it should be criminal to sell these type rackets to a beginner especially children.
3. Buy only Red / Black rubber coverings not red/ red or black/black or blue/green green/yellow etc etc)
If you are 50 or older and had played with pimpled rubbers all your life and want to stick with it you may do so but keep in mind that you will still be bringing a knife (pimpled racket with no sponge) to a duel where your opponent has a nuclear tipped missile (smooth inverted rubber with carbon or composite layered blade, fully glued up to the max.....you will understand gluing up, blade etc later) .
Many basement players do not want to flat out give up their pimpled rubber rackets for the short-term gain of beating up on other clueless basement level players. But you must look at the long term gain and live with the short term humiliation of losing to players (especially your mother-in-law or father-in-law LOL ) if you want any chance to play with the big boys and girls of competitive table-tennis. Microscopic changes you make to your racket greatly effect your play even using same type rubber. Playing with a racket with 1 mm sponge smooth rubber is drastically different from using a racket with a 1.7 mm sponge smooth rubber and it may take weeks to months depending on your talent level. That said, adjusting from a basement racket (pimpled out or pips out ) to a competitive table-tennis racket takes at least the same amount of time & it won't happen in one day. In fact, conversely, many competitive players (most of whom use only smooth rubbers) try somewhat secretly, switching to pimpled out rubbers and when they see that it won't happen in a day or week, they go back to smooth rubber and bad mouthing pimpled out rubber players. ( These smooth rubber players are known by names such as closet junkies, robotNazis etc )
Balls must be 3 star or platinum only. ( 2 star or gold quality balls can be purchased but for training & drills only) . 1 star (Silver) are not recommended. Even in 3 star balls, many brand names are of poor quality. Recommended brand names are Nittaku, Butterfly, Double Fish, Gambler, Double Happiness & Friendship.
As far as shoes go, you can start with volleyball or tennis shoes (litest models) and then move on to table-tennis shoes. Being one of the fastest sport (on feet) , even tennis or volleyball shoes may be too heavy, not to mention basketball and cross-training shoes which will be way too heavy.