What should be the budget to start?
You don’t need any special budget. Probably allocating 10% to 20% of your total savings to silver (and if you can afford it) gold bullion coins seems like a good idea for a passive investment. Gold takes up a lot less space. My personal feeling is that market manipulation of “paper gold & silver” will have to result in prices triple what they are today. The question is not “if” but rather, “when.” It will come in no more than 15 years. But don’t buy from coin shops where they may mark a coin up to double or triple the “melt price” Never buy “polished “ defaced, or grossly overpriced “commemoratives” from “private mints.” Study the market a little bit and you should be able to buy nice new looking sovereign coins from Britain, Canada, Austria, etc. for the same price as the metal. Beware of China Pandas as there are many gold plated counterfeits around filled with tungsten.
Like anything else, you need to know what you are doing — and don’t believe the hype you get from dealers. Always consider the source of your info.
I am not a dealer and have no axe to grind. I feel that your own business is always gong to be a better investment of your time & money than any passive investment.