If I buy a share from myself at a higher price, will that drive the price up so I can sell all my shares the higher price?

No, this isn't possible, especially not when you're trading a highly liquid stock like Apple. When you put in your buy order at $210, any other traders that have open limit sell orders with the correct parameters, e.g. price and volume, will have their order(s) filled. This will occur before you can put in your own sell order and purchase your own shares because the other orders are listed on the order book first.

In the US, many tax-sheltered accounts like IRA's have specific rules against self-dealing, which includes buying and selling assets with yourself, so such a transaction would be prohibited by definition. Although I'm not entirely sure if this applies to stocks, the limitation described in the first paragraph still applies regardless.

If this were possible, rest assured that high-frequency traders would take advantage of this tactic to manipulate share prices. (I've heard critics say that this does occur, but I haven't researched it myself or seen any data about it)

This probably won't be a popular answer due to the many number of disadvantaged market participants out there but:

Yes, it is possible to distort the markets for securities this way. But it is more useful to understand how this works for any market (since it is illegal in securities markets where company shares are involves).

Since you asked about the company Apple, you should be aware this is a form of market manipulation and is illegal... when dealing with securities. In any supply and demand market this is possible especially during periods when other market participants are not prevalent.

Now the way to do this usually involves having multiple accounts you control, where you are acting as multiple market participants with different brokers etc. The most crafty ways to do with involve shell companies w/ brokerage accounts but this is usually to mask illegal behavior In the securities markets where there are consequences for manipulating the shares of securities. In other markets this is not necessary because there is no authority prohibiting this kind of trading behavior.

Account B buys from Account A, account A buys from Account B, etc. The biggest issue is getting all of the accounts capitalized initially. The third issue is then actually being able to make a profit from doing this at all. Because eventually one of your accounts will have all of the shares or whatever, and there would still be no way to sell them because there are no other market participants to sell to, since you were the only one moving the price.

Therefore this kind of market manipulation is coupled with "promotions" to attract liquidity to a financial product. (NOTE the mere fact of a promotion does not mean that illegal trading behavior is occurring, but it does usually mean that someone else is selling into the liquidity)

Another way to make this kind of trading behavior profitable is via the derivatives market. Options contracts are priced solely by the trading price of the underlying asset, so even if your multiple account trading could only at best break even when you sell your final holdings (basically resetting the price to where it was because you started distorting it), this is fine because your real trade is in the options market.

Lets say Apple was trading at $200 , the options contract at the $200 strike is a call trading at $1 with no intrinsic value. You can buy to open several thousand of the $200 strike without distorting the shares market at all, then in the shares market you bid up Apple to $210, now your options contract is trading at $11 with $10 of intrinsic value, so you just made 1000% gain and are able to sell to close those call options. Then you unwind the rest of your trade and sell your $210 apple shares, probably for $200 or $198 or less (because there are few market participants that actually valued the shares for that high, the real bidders are at $200 and lower). This is hardly a discreet thing to do, so like I mentioned before, this is illegal in markets where actual company shares are involved and should not be attempted in stock markets but other markets won't have the same prohibitions, this is a general inefficiency in capital markets in general and certain derivatives pricing formulas.

It is important to understand these things if you plan to participate in markets that claim to be fair. There is nothing novel about this sort of thing, and it is just a problem of allocating enough capital to do so.

Yes it is possible but with a caveat. It is a pattern that can be observed in many lightly traded stocks that usually have a small market cap. I am talking about a stock that trades less than 2,000 shares per day on average.

The market maker will always compare the highest bid and the lowest ask. A trade will happen if the highest bid is at least as high as the lowest ask. Adding one share (or a million shares) at a higher asking price, here: $210 instead of $200, will not have any effect at all. Nobody will buy the share.

Adding a bid for one share (or a million shares) at a higher bid price will trigger a sale. If you bid $210 for one share, you will pay $210 for one of the shares that were offered at $200. If you have $210 million in cash and add a bid for 1,000,000 AAPL at $210, you will pay $210 for all shares with an ask of $200.00, then $200.01, then $200.02 until you either bought all shares with an ask up to $210, or until you bought a million shares.

With AAPL, you probably bid the price up to $201 with a million shares, so you made lots of people very happy while losing about 10 million dollars. So let's say this is a much smaller company.

You have driven the share price up to $210, but there is nobody else bidding above $200. So nobody is going to buy your shares. Until some people think there is something going on and enter higher bids, but then some people will take advantage of this and ask lower than your $210. And there will be more people trying to make cash by selling their shares at a good price than people tricked into bidding over $200, so it is most likely that you lose out.

(This completely ignores legality; attempting to do this would be market manipulation and in many countries illegal. I don't know if losing money in the process would protect you from criminal charges).