Knowing as much or less then you do about stock, I would talk to an accountant. Of course a broker will want you to do something that will result in them getting a hefty commision, but an accountant will make sure you don’t wind up paying a ton in taxes on it.
If your (or your father’s) bank has brokerage (trading) accounts available, that’s probably the easiest option – you might not have to open a new account, and it will be local in any case. Assuming that you’re not going to become a frequent trader, the price of commissions and fees aren’t likely to be as big deal for you as convenience. If you are, shop around a little more for low fees. If your bank doesn’t have investment accounts, look around for another one - Bank of America, for example, just announced free trading (no cost for trades) for account holders with their investment accounts and $25K or more of assets. (Note: I don’t use them, and I’m not legally able to recommend them. Consider this an example. I am NOT a financial planner or any other financial professional; I’m a computer programmer.)
Once you’ve found your broker, call them and ask what to do with the certificates. I’m not sure how common it is to actually hold stock certificates any more: certainly I don’t have any, and I own a fair amount of stock. The brokerage firm holds the actual certificates (assuming they exist at all as anything other than digital signals). I suspect they’ll offer to take them off your hands and turn them into something electronic.
Once you’ve got the certificates with an account, just tell them you want to sell (assuming that’s what you do want).
Is it worth it to you to sell? That’s where you start to get into the financial advice thing. None of us are qualified to tell you. Go find an independent financial advisor (one that doesn’t actually buy and sell stocks for clients) and pay for a hour or two of their time. In fact, do this first!
Get a financial advisor! To sell or hold an amount of stock that large is too sensitive a decision to leave to the teeming masses and the 1,000 different opinions you will receive. You have not only a decision about future prospects for the companies but also tax implications and then decisions on where to invest the cash if you liquidate. And you’ll need a professional opinion from someone with whom you are willing to share personal financial details.
If you choose not to do that, go out to the company website and look for the “Investor Relations” Link. There should be an address or phone number for the company’s stock registar. Get on the blower with them and find out where to send the certificates if you choose to sell the stock.
I can’t address the tax consequences or the matter of whether to hold or sell, but as a brokerage operations veteran, I can offer some advice on the disposition of the actual shares. Your first step should be determining the validity and value of the certificates themselves - you don’t mention their age, but it’s quite possible the paper itself is worthless, given the possibility of escheatment or a prior loss claim. There is also the possibility the stock has been subject to any and all manner of reorganizations (forward splits, reverse splits, etc.), all the more likely for “petroleum based” companies.
Starting with the company’s Investor Relations is not a bad idea; their designated transfer agent (typically a bank such as Bank of New York or Citibank, etc.) should be able to verify if the certificates are outstanding, and advise if they have been subject to any splits or reorgs. Alternatively, you can send the certificates to your financial services advisor (i.e. your Broker), whose first step will be to send the certificates to the transfer agent themselves, and has the added benefit of getting your shares into “street name” - the industry term for the “digital signals” referenced earlier, which will make it immeasurably easier to sell if and when you decide to do so. (As an aside, brokerage firms do not typically hold stock certificates either; most stock is held in a central depository, which facilitates transactions for most all Wall Street firms.) More specific information would require more information, such as the name of the company (stock symbol?) and the date(s) the certificate(s) were issued.
And yes, you will need an account with a brokerage service to sell the stock if you decide to do so; individual investors cannot just sell stock on the market. However, many banks offer investment accounts nowadays, and there are always discount brokers.