Hi KumquatMay congratulations on considering adoption, it is exciting and life changing.
In terms of your budget I would start with what you spend on the two of you and work it all out onto a spreadsheet; be honest, include things you spend on now. If later you decide to cut back on this or that, so be it, you can alter it, but to begin with put everything in. We have done this by having a notebook and writing in everything we spend over a period of time.
In terms of clothes etc you don't buy new clothes all the time but you do buy something annually (usually) so you may be able to work out roughly what you need per month to put aside for when you do buy clothes. Likewise a new car is a rare outlay for us (I mean new to us, we always buy second-hand) but we put aside a bit each month for a car when we will need it and for holidays etc.
So start with what you need for you two, which you may or may not have done.
Personally, to add in the cost of two children the best thing to do would be to budget for the children as best you can, again looking to see what they may cost per month. This will be the best way to work out what your costs would be. Others will be unlikely to be able to tell you because we all probably spend different amounts on different things. So people who are great cooks, producing amazing meals from their home grown veg and whipping up cakes for pennies, may spend less then me! My kids eat a lot of pizza, chicken nuggets too, which are not cheap but are often all they will eat (please do not judge me 
) and they are huge fans of
these over priced cake bars!
To work out costs for clothes and equipment, I would do the same as you do for your clothes and car. You can get given stuff, that does happen, but not always, and so you may want to investigative costs of second-hand equipment etc and allow for some new things too, (Mattresses for kids should be new).
Do some window shopping and see how much money you would need to spend on general outlay and on ongoing costs in your area. Friends and family will often offer to buy or contribute to big items and toys etc but it is the day to day costs that need to be accounted for too.
Our family has gone up by one, from three to four (by adoption), and I have noticed we are getting through more food and doing more laundry, it's obvious of course. However, if you had asked me before if I would really notice it, I would have said something like "Kids are small, they don't eat much, their clothes are small etc!" BUT you do notice it. Their shoes are very expensive and they grow fast, they grow out of clothes quickly when little.
So they are 100% worth any costs or expenses, and in fact very quickly they become the very centre of your world, but to know how much you would need to support them you will need to do some maths work in your area. Good luck.