The answer is not so clear cut as you may like, but all depends on your aversion to risk and your financial well-being.
At both $20,000 and $200,000 you can invest in metals, it just depends on what you're willing to lose. That's right, even if metals are pretty non-volatile, when you invest money you need to be prepared to take a loss. If losing major portions of your initial investment (i.e. $20,000 or $200,000) would cause you financial distress then I would suggest not investing that money.
If you are looking to trade in the United States, the American Metals Market looks like a good place to start. There is also the APMEX and the American Metal XChange.
In london there is the London Metals exchange, a comprehensive list of fees, pricing, membership and FAQ's are located HERE on another one of my answers.
To answer your question directly, to invest in metals you can go through popular exchanges like the ones I've linked, or you can try to find local traders. In the end you will probably need some sort of storage, as you are buying and selling physical items. Sometimes you can find places that will keep metals for you for a small fee.
If you are looking to make money off of the rise in these commodity prices, I would suggest investing in mining companies (even though I know you said you didn't want to). They are advantageous in a few ways:
1) You have no physical property to keep safe and secured, and no shipping charges to worry about.
2) Trading shares is much more easily and readily done than trading physical materials. You need to think about the liquidity of your assets.
3) Mining stocks are found in every market around the world, making it easy to invest where you are.
I hope this helps!