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Separate stain and varnish is one approach, that's usually the approach favoured by the pros. Problem is that stain on it's own is very difficult to adjust on the first coat, it really soaks into the grain. In that case you need to very carefully blend or choose your base stain and apply it with some confidence then look at where you are. You can adjust (except making it lighter) from that point with stain in varnish to the point where you are using only clear varnish. This is really only for fine furniture, things like floors are a slightly different technique.
If you are doing multiple doors for example you need to be very careful to get a similar technique on each door, that's complicated.
For fine furniture there's also something called grain filler which allows you to get a piano gloss finish, that also makes dye/varnish application easier in some, respects harder in others.
It's such a huge subject and a lost art in some ways. Contemporary joiners are keen on spraying because it makes it easier to get consistency on the large areas that are a requirement of modern furniture.
I might be moving away from describing what you are trying to achieve but a lot of the techniques are covered by French polishing where you use a 'rubber'. I don't know why the subject is so ripe for double entendre :)
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All of that above and I didn't really address the question!
It shouldn't be beading up, water based should stick to water based. You might need to key the surface with a very fine (250 grit or above) sandpaper. If you let me know the brand and product that you're using I might be able to help a bit more.
Aside from the tips that have already been mentioned it's not an easy thing to do. It's just the more experience you have the less likely you are to end up with a streaky mess. As withered preacher mentioned you should be applying thin coats with cloths. I like mutton cloth for this job. Any brush application will be more likely to overload the surface in some areas, there are foam brushes which some people like.
You are not applying much at all but slowly darkening the wood in as even a manner as possible. Make sure you don't go over your work once it's started to dry until it's fully dry that way you should be able to take an entire coat off if it goes tits up by using a wet cloth (before it's dried). Don't let it dry if you think it looks streaky. Normally it drys very quickly. If you are watering the dye down to make it easier to work with it will take longer to dry and it needs to be very well mixed. I found wetting the surface made it worse, that might just be my technique though.
It's then equally difficult to know when you should apply the varnish especially if you don't have any test areas as it will change the depth of colour.
Sometimes you tend to judge the results too critically as well. I'm sure there are a lot of not so even professional jobs if you shine a very bright light on them. Just bear in mind it's a craft, you can't always avoid the human element with a hand polish, sometimes it still ends happily :)