Ok and oldie but goodie question; here I go:
1) Quality is a process, is taking care on each step of it.
In a case of resampling an image there is no "quality" loss, (except if you make mermelade of your own photo, probably compressing it like hell) What you have is information loss when you downsample it. You have a "information guess" when you upsample it.
2) There is no CSI program that perform miracles in the terms you need (However, I have seeing some forensic image processing program that fairly shows a licence plate from a very low resolution image, or from a very narrow angle. The result was ugly, but you could clearly see the licence plate.)
3) So, the programs use diferent "guess" methods to try to asign information to the new pixels.
Some real tests
Here is a controlled exercise of resampling. Reset your browsers zoom so you see them in real size.
a) In the center are 3 images. The reference image is the one marked 100%
b) Next to it there are some smaller marked as 300% and 200%. They are 1/3 and 1/2 respectively, when they are upscaled they will have a resampled ratio as marked.
The programs used (in order of appearance):
1) Just scaled (this is the so popular "pixelation", the technical name is "nearest neighbour") 2) Irfan view Lanczos Filter
3) PhotoZoom 4) Reshade
Photo: Scott F. Snyder
Model: Amy Lee Fathbruckner
The normal Photoshop bicubic filter looks simmilar to Lanczos Filter.
This is an oldie test, there is a chance the programs now make a better result, but do not expect a quantum leap.
They have no aditional process. You can add a small sharpening after the initial resampling. Photoshop has one marked as bicubic sharper.
I would resample in round numbers. Not 150%, not 234.567%
If you simply strech it inside Ilustrator or simmilar, the result will be like sample No. 1.
In my opinion, a 200% upsampling is acceptable. A 300% just in case of backgrounds.