Regarding your own accounting system, the 3 goals you mention kind of require different answers to your question.
Before we get to that, there are two main things to keep in mind: the utility an object has to you, and its market value. And to get to the punchline early: some things appreciate (potentially cars, also art, etc..). So if you're tracking for market value (which is probably relevant to net worth), you'd need to keep that in mind.
I'm going to leave the question of net worth at that for now, as others are speaking to that, and it's not really my area (unless you have any other questions).
Regarding your other two goals, for budgeting there's a further detail: cash budgeting, or budgeting on 'accrual' basis (which is in line with income/expenses that are more accurate).
Cash budgeting would only take in to account the amounts spent or received. I'm going to ignore that as well.
So let's talk about recording assets you purchase, from purchase date until the date you sell or discard them. This will be in terms of your internal income/expense tracking, and budgeting (accrual basis).
The biggest factor is the time period during which you're going to use something up. Since you're going to totally use up this week's groceries within the month, you expense those.
The car you buy to use to the next 3 years, you would book in your system as an asset (not an expense), and you can then depreciate it each month - either as 1/36 of the cost, or a more elaborate method. Alternate method 1: your use of the car.. if you could estimate how many total miles you plan to drive on that vehicle, and if you track miles driven each month, you could determine a purchase amount per mile driven, and each month book that.
(that entry would be debit vehicle depreciation/expense), and a credit to accumulated depreciation (contra asset to the car asset).
The difference between the car asset (original purchase price) and accumulated depreciation (your use of it) is your car book value.. if/when you sell it, the difference between your sale price and the book value would be revenue (or loss).
And once you have an idea of vehicle cost per month, that can be a non-cash expense in your budget (if you're using an accrual-based budget). This could apply to a bicycle/motorcycle etc.. as well.
Clothing - jewelry could certainly be booked as an asset if the value is that high, and/or footwear, other particular items. But yes otherwise most would simply expense those purchases (not because each clothing item is going to wear out in a year, but because keeping an asset for clothing would be very high-detail).
Computer systems, audio systems etc.. could all be treated same as the car, you can set depreciation across the period of time you expect to own them, or some other creative basis if you prefer.
Other small household items, furniture, etc.. most would find the time it takes to 'book' those (and track usage/remaining life) not worth it, but your decision can factor in a particular piece or two or particular unique items.
Each thing you book as an asset, to keep your system intact you'll need to be sure and take it off your books when you sell or discard it.
Circling back to net worth, only the biggest assets would figure in (besides actual monetary items like checking/savings/investments/retirement funds). There's usually a line item for 'household possessions' I think, where everything else would be included (car, bike, computer and audio etc..). But that doesn't have to interrupt your own internal tracking process for your purpose.
And again if you own something that might appreciate in value, you'd want to adjust accordingly - probably wouldn't include in your budget or income, but it could be something to include in your net worth statement.