Something to note is that when a company announces a share buyback program there is usually a time frame and amount of shares that are important details as it isn't like the company will make one big buy back of stock generally. Rather it may take months or even years as noted in the Wikipedia article about share repurchases.
Wikipedia covers some of the technical details here but to give a specific set of answers:
When a company announces a share buyback program, who do they actually
buy back the shares from?
From the Wikipedia link:
"Under US corporate law there are five primary methods of stock repurchase: open market, private negotiations, repurchase 'put' rights, and two variants of self-tender repurchase: a fixed price tender offer and a Dutch auction."
Thus, there are open market and a couple of other possibilities.
Openly traded shares on a stock exchange?
Possibly, though there are other options.
Is there a fixed price that
they buy back at?
Sometimes. I'd think a "fixed price tender offer" would imply a fixed price though the open market way may take various prices.
If I own shares in that company, can I get them to buy back my shares?
Selective Buy-Backs is noted in Wikipedia as:
"In broad terms, a selective buy-back is one in which identical offers are not made to every shareholder, for example, if offers are made to only some of the shareholders in the company. In the US, no special shareholder approval of a selective buy-back is required. In the UK, the scheme must first be approved by all shareholders, or by a special resolution (requiring a 75% majority) of the members in which no vote is cast by selling shareholders or their associates. Selling shareholders may not vote in favor of a special resolution to approve a selective buy-back. The notice to shareholders convening the meeting to vote on a selective buy-back must include a statement setting out all material information that is relevant to the proposal, although it is not necessary for the company to provide information already disclosed to the shareholders, if that would be unreasonable."
Thus it is possible, though how probable is another question.
While not in the question, something to consider is how the buybacks can be done as a result of offsetting the dilution of employees who have stock options that may exercise them and spread the earnings over more shares, but this is more on understanding the employee stock option scenario that various big companies use when it comes to giving employees an incentive to help the stock price.