I used to manage small business property/casualty accounts, in my past life, and I can tell you that they are right, you will need an entirely separate policy. I can also tell you that there is no way to determine what "enough" insurance is. However, as someone stated - if you're a sole proprietor, your personal assets could be tampered with if your claim exceeds your limits on your business insurance.
Typical liability coverage for small business starts at $1,000,000 per claim, $2,000,000 per aggregate (year). I wouldn't sell anything less than that. Those limits are going to be your most costly, so I always recommend getting an additional quote for $2,000,000/$4,000,000. In some cases higher limits cost under $10 a policy term because some companies lower rates for insureds who value good coverage.
You may also find that your personal policy won't cover your equipment because it is used in your business. Insurance companies spend gazillions of dollars hiring people who will find out the difference between what you use for your business and what you don't, in the event of a claim. I would recommend looking into a Business Owners policy, which is a small package that can include property coverage and liability in the same policy at a really affordable price. The annual fee for this policy starts at $500 per year for the basics. It also generally includes a coverage for Business Income expenses which helps in various situations should something occur that disables you from running your business. It doesn't protect against your personal illness or anything of that nature but should something happen to your home and you're unable to work, it can help you rent another kitchen, pay business expenses, etc. while repairs are made.
The only other thing I would note is that if you're making deliveries from a personal vehicle, your standard auto policy will not cover you. You would need to contact your insurance company and request an endorsement to add business use to your vehicle (it's not always available with personal insurance carriers) or get a quote for commercial auto insurance. I also recommend getting commercial auto because it will still allow for personal use of the vehicle and the limits also generally start at $1,000,000 and the premiums are comparable.
If you're looking to deviate from your personal insurer for a commercial quote, I'd recommend Travelers Insurance Co. or Safeco Insurance Co. Travelers seemed to have great small business pricing, but Safeco had been developing a lot of "at home" business classes, specialized for people working from home. You may find some unique coverages there that give you more bang for your buck.