Y'all make it sound like 304 is some really bad stuff to work with. And maybe the stuff you have is, I've heard some folks who got imported material that gave them a lot of grief. But normally 304 really isn't that bad. And it has much better corrosion resistance than 400 series material.
There is no place for carbide in this application. Quality USA made HSS or cobalt HSS is the key. This is a good application for a 135 split point, though I've used 118 for this without problem. I use Chicago Latrobe for my general purpose drills. They're not expensive and they hold up very well.
The biggest problem most of you guys have is underfeeding it and then trying to cut with a dull cutter. Don't underfeed and don't try to cut stainless with a dull tool you'll only work harden it.
The key to success is a quality tool ($2.00 USA HSS drill bit) and to feed it right. It has a narrow window compared to other materials. You can push a 3/32 drill through aluminum at 2 inches per minute up to 40 inches per minute, no problem. Wide window. Stainless has a narrow window. If you will turn your 3/32 drill at 1200 RPM and feed it at 2 inches per minute you will have no problems. If you underfeed it, it will rub instead of cut which will quickly dull the drill and harden the work piece. That bears repeating: If you underfeed it, it will rub instead of cut which will quickly dull the drill and harden the work piece.
Use a cutting fluid if it makes you happy but it really doesn't do a whole lot drilling shallow holes.
The chips will come out in a curly cue as a bare minimum to indicate you're feeding it hard enough. I generally feed it hard enough the chip breaks and I get short thick fat little chips. If you're squeaking and getting a birds nest around your drill you're not feeding hard enough.
Once the drill is dull, stop. 3/32, throw it away. Fed properly you can get hundreds of holes. Fed improperly it can dull in just a couple.
Other than underfeeding, the next big thing people run into is overfeeding. This generally happens once the web of the drill exits the back of the work piece and you lose the majority of the resistance to feeding and you overdo it. So be careful once the drill starts to exit the back side of the work piece, otherwise it is easy to wipe out the corners of your drill.