What should I include in my asset calculations? How do I know what is an asset and what is a liability?
It's an asset if you can sell it or if it provides cash flow. For example, if you own a house, the house is an asset, even if you have a mortgage, because you can sell it. If you rent your home from someone else, it's not an asset, because you don't own it and can't sell it. Stocks, cars that you own (not lease), cash, checking and savings accounts, precious metals, collectibles, and so on are assets.
Liabilities are debts; that is, what you owe. So, for example, liabilities include: balances on mortgages, student loans, credit cards, other loans, any tax or judgment debts, and so on.
There are a number of 'net worth' approaches you can use.
The broadest is to add up all retirement accounts, savings accounts, insurance, home value, and value of possessions that can be sold for a price worth listing, cars, collectibles, art, etc all included. Then subtract any liabilities, mortgage, credit card or personal debts, etc. This is an "inheritance" net worth, everything you'd leave to your heirs if you die.
Many people are looking for their net worth to track their savings to retirement. This starts with the above, but I'd remove cars and possessions. Unless you have an expensive piece of art to fund retirent, these all should be subtracted (or not counted in the first place). I'd also not include the house at all. At retirement, you need to live somewhere and shouldn't count on a big enough downsize to offer too much cash. This is my approach, my opinion. When I see others post net worth, I often see someone worth say $250k but they live in a $300k house. They really have a house but no other assets. Again, this is my approach.