In theory, any knife with a flipper tab should open by finger movement alone. Of course, this is reality. Reliable opening mostly depends on your technique, the detent, and the action. You can usually get by with two out of three and a lot of one can compensate for less of another. Really though, you should have all three. You want a knife that moves smoothly and without “grit”. You want a detent strong enough to keep the knife closed in your pocket and ideally strong enough that you can’t shake it out by wrist movement alone. (You also want it to lock securely and without blade play when open.) Then you’ll want to practice your technique by flipping it open and closed all day long. 
The real question is “how budget do you need to go”. In general, getting all of those things together can get harder the cheaper you go. As you get down in price, there can be compromises in both materials and quality control. The latter can make ordering knives a bit like playing the lottery. I’ve owned a lot of budget knives over the decades, from M-Tech to KKWolf. These days, Ganzo is the cheapest I’ll go.
I’ve had pretty good luck with Ganzo. My FH11 had a smooth action and almost perfect detent from the factory. It locks up sturdier than a lot of Kershaws I’ve owned too. It’s been one of my favorite beaters for a while now. On the other hand, the FH41 I got recently had an extremely weak detent so who knows. Speaking of Kershaw, they generally do a good job in the $20-30 range and they’ve got some decent manual knives like the Fraxion and Atmos. Something that sets Kershaw above companies like Ganzo is great customer service. If you’ve got a problem, call them. They’ll help you!
The best advice I can give you is to consider saving up to the $40-50 range. This is where things really start to get nice and some incredible deals start to happen. Civivi has a line of knives using G10 scales, ball-bearing actions, good blade geometry, and properly heat-treated 9Cr18Mov for $42.50. Having owned, used, carried, disassembled, and tinkered with a few of them now; I can tell you that the quality is consistently amazing. I rarely see a fit and finish that good on knives that cost twice as much. Another good brand to check out is Bestech. Their base line is usually $52 but several models have been recently dropped down to $39.99 on various sites. Check out the Spike, Grampus, Thorn, or 12C27 versions of the Beluga.