There are a lot of items here and you could get carried away and spend a lot, but not all of these items are essential, so be careful with your money. You'll generally build up your kit collection over many years of playing the guitar.
If you're from the USA, take a look at my Amazon Store for some ideas.
Some starter guitars for children.
There are cheaper guitars, but the Fender Starcaster and the Epiphone Les Paul are solid choices for beginners. I wouldn't recommend going any cheaper than the Starcaster as the quality starts to decline - it's very difficult to learn to play when the instrument keeps going out of tune.
Roland has some very good amps in its Cube range, but if you're looking for a cheap amp to practise with, go for the Marshall MG10 CF.
There are also some good packages available for beginners. Left-handed guitars (last in list below) are also available but tend to be slightly more expensive. If you're looking for a guitar for a young player of 8 or 10, the Stagg is a nice starter guitar.
Money no object? Then treat yourself to the best!
If you already have an amp that does not have any built in effects, you could use a pedal or two. Here are some youtube videos demonstrating the pedals: Boss DS-1 Distortion, Boss Ch-1 Super Chorus, Digitech RP55 and Digitech XDD DigiDelay
Of course, there will be broken strings, so make sure you have spares. I use Ernie Ball's Super Slinky strings for electric guitar. For acoustic guitar, Martin and D'Addario are good choices.
I recommend purchasing the 6 pack of Tortex plectrums. Experiment to find the one that suits you. Personally, I prefer the orange 60mm, but give them all a try.
You'll need a gig bag to transport your guitar and a lead to connect to the amp. Generally, the more expensive the lead the better the sound quality (but this is not always the case, so always read the reviews).
Look after your guitar. Lemon oil is used to treat the fretboard. If your guitar gets really grimy after all that practising, use some polish and a microfiber cloth to clean the body. You may want to invest in a guitar care kit.
If you are learning from a book, a music stand is very handy, invest in a sturdy stand. You'll also need a guitar strap - a cheap one is good enough for practising, but if you're going to be standing up and performing you'd better invest in a higher quality strap.
If an amp is too expensive you could try a headphones amp. A good multitool is useful. A mic stand pick holder is ideal for singer guitarists. Don't let that top ten hit get away - record your ideas with a digital recorder.
If you want to play Wonderwall or Hotel California a capo is essential. Although not compulsory, a guitar stand is very convenient. There may be a time when your tuner is out of batteries, a tuning fork in E will help you manually tune your guitar. Finally, if there's one effect pedal every guitarist should have, it's a wah pedal - trust me!