Regarding climb milling: it depends entirely on what sort of machine you have. If you're running a converted Bridgeport or some really cheap machine, you might not have ballscrews, and thus doing climb milling is a bad idea (good info is
here. In general, CNC Cookbook is a pretty good learning site.) Stick with conventional milling, and don't use carbide, it hates conventional cutting. If it's got ballscrews, definitely climb milling.
I'd personally second the call for a bigger end mill, if possible (remember, rigidity increases with the fourth power of diameter for cylinders. It's a reasonable approximation. Switching to a 1/4" will get you three times the rigidity.) That being said, you can get a 3/16" with 7/16" length of cut from McMaster, which isn't terrible. 2.5xD stickout. Shouldn't really be any big problems with that. If I'm recalling correctly, carbide has about 2.5x the elastic modulus of HSS. So it's roughly equivalent to the 1/4".
One thing to note: if you're going to be slotting at any point while cutting, you'll need to turn down the parameters. Slotting is very hard on the tool. Start by reducing the feed and speed 20%, and make sure you're dumping on coolant. For plunging, try 1/5 to 1/7 of your XY feed. End mills hate plunging.
I'd advise running some test cuts on some scrap material. Get a feel for how it cuts. Don't worry about breaking the cutter, just be ready to E-Stop the machine. You're playing with $13 of endmill (McMaster.) Tooling is disposable. If it doesn't take the cut, switch to a carbide or 1/4".
For CAM programs, try
HSMXpress. It's free, and while I haven't used it, it looks pretty user friendly, and is totally integrated into SolidWorks.