Q: What is the “best” exercise / routine to do, and are there exercises that burn more calories and are therefore better for fat loss?

A: The “best” exercises are often the most simple and basic ones (e.g. deadlift, squat, press, pull, carry, plank) because they represent the fullest expression of human performance under load, and because they require strict focus and concentration for developing efficiency and progression in your movement skills.  

Simply put, the better you can get at a task, the more intensely you may perform that task, the more it challenges you and the more return on investment you get.

Fat loss is dependent on a balance of how much fat you store vs. how much you expend (notice how I didn’t say “burn”) over a long enough timeline to see appreciable loss.  How much energy you expend is a combination of several mechanisms: your basic daily energy requirements (BMR), your non-exercise activity (NEAT), the thermic effect or digestive requirements of food (TEF/DIT), and finally exercise or work (EA).

BMR + NEAT + TEF + EA = Energy Out

If all these add up to more than the food energy you take in, you will lose fat.  Some exercises burn more calories per minute on average than others - but of course this is dependent on how much effort you put into the exercise.  No matter what exercise you do, the harder or faster you work, the more calories per unit time you burn.  
If you perform kettlebell swings with a challenging weight, and burn 20 calories per minute, then that is very effective for fat loss… but not if you only do 20 seconds worth for 8 rounds and call it a day.  Yes, it is hard!  But you only worked for 21/2 minutes total.  That’s only 52 calories!  If you eat at maintenance calories (the amount to sustain weight), you need to burn at least 300 calories per day to see appreciable and consistent weight loss, until that loss slows of course. 
You may have also been told that high-intensity or “HIIT” training is great for fat loss because of EPOC (Excess Post-Exercise Oxygen Consumption).  The American Council on Exercise describes EPOC as “the amount of oxygen required to restore your body to its normal, resting level of metabolic function (called homeostasis).”  Following intense exercise an “oxygen debt” accumulates, and it has been reported that EE may remain elevated for a few minutes up to 8 h post-exercise depending upon the nature of the activity (e.g., intensity and duration).  Thus it is argued that HIIT can help you lose weight because you are burning more calories than if you were not in oxygen debt.  Unfortunately this logic suffers from the same practical flaws as the ketogenic diet and intermittent fasting.  You can expend more calories at rest than normal, but if you eat more calories than your oxygen debt accounted for, you will not lose weight.

The rate of energy expenditure at a single point in time does not dictate end of day energy balance.

As a matter of fact, several recently published meta-analyses focused on comparisons of exercise protocols have not supported the potentially beneficial effects of high-intensity interval training on long-term fat mass reduction compared with continuous exercise.
What we also observe is that subjects who undergo intense training protocols typically move less for the rest of the day because they are so exhausted and mentally fatigued, thus offsetting the mild EPOC effect that occurs by a reduction in TDEE.  EPOC is such a mild effect, that the documented difference in EE, accounting for baseline EE, is about 32 kcals per day for H.I.I.T.†  Ten-minutes of brisk walking could accomplish the same thing.
It’s the total calories burned by the end of the day that matters, not what exercise burns the most per minute.  Unless your goal is to only lose 1 pound, you need to do the exercise routine that you can stick to for the minimum amount of time necessary for that fat loss to happen.  If your goal is 10 lbs, your minimum timeline is 5-6 weeks, and that’s if you do everything right (i.e. food tracking, not missing workouts, getting enough protein, maintaining an adequate calorie deficit, etc.)!

† Panissa, VLG, Fukuda, DH, Staibano, V, Marques, M, Franchini, E. Magnitude and duration of excess of post-exercise oxygen consumption between high-intensity interval and moderate-intensity continuous exercise: A systematic review. Obesity Reviews. 2021; 22:e13099. https://doi.org/10.1111/obr.13099