What is the minimum cost to start mining in a brand new condition?
The barrier to entry can be as low as $400 if you buy a single RTX 3060 video card for altcoin mining. Avoid 'cloud mining' contracts, since they will either be a scam or you'll be locked into hardware that can become unprofitable. The cloud provider, even if they're honest, can't adapt to market conditions.
Another alternative if you're in the U.S. is to buy stock market shares of mining farms, like Riot Blockchain. But the problem is double taxation, since the company has to pay 21% if it makes profit, then you have to pay 15 or 20% on top of that.
The cheapest price for a brand new ASIC is probably $3000-6000. The higher your power cost is, the newer the model of ASIC you should get. Be wary of sellers that offer a price cheap enough that the payback period is < 300 days. It's probably a scam.
Use
this site to see which ASIC models are the best deal at your energy cost. Older, less efficient miners are higher risk if coin prices go down, but higher reward if coin prices go up.
And can I do mining on my personal computer or do I have to have a separate mining station for this?
If you have a gaming PC, you can use your existing video card, or add another video card. But I recommend building a dedicated mining rig with 3 to 10 GPUs.
When it comes to ASIC miners, you just need to connect an Ethernet cable and power cord. The ASIC usually has its own onboard computer
How reasonable would it be to mine in countries where mining has not yet been legalized?
The smaller your mining operation, the less likely you are to get caught. What do you mean by 'not yet legalized'? Most countries either do not regulate crypto mining, and a few outright forbid it (like Communist China). Most nations treat mining as a regular business with regular tax laws. Here in the U.S., the IRS treats coins as regular income that is taxable the moment it's deposited into a wallet.