I get this question all the time. Somehow people are often drawn to the sound of the saxophone. Despite only being only invented in the 1840s, it has been prominently featured in almost every genre of music since its inception. This could have given rise to its visibility and popularity. Or it could be the fact that when played well, it can sound a lot like the human voice.
I learnt the piano as my first instrument, so if I was comparing the saxophone to the piano, the saxophone would seem much easier to learn. Firstly, on the saxophone you would only play one note at a time. You would only need to read the treble clef. Also, you do not need your feet to step on any pedals.
As with learning anything new, there will be a sense of challenge. If you had prior music experience before, it will probably speed up your learning time. I would say with consistent practice (at least 3 times a week), most students will be able to play a simple song in 2-3 lessons and a pop song in 4-6 lessons. Of course there will be exceptions.
Having a private tutor will definitely make the process faster, as well as ensuring you do not make any bad mistakes. If there are budget constraints, the bare minimum would be to at least have your first lesson with a teacher. If possible, go for at least 4 lessons before trying to practice on your own with online resources. The thing about online video lessons is you do not get any feedback about your playing. So it is always good to check in with a teacher personally. Some online studios or teachers accept video submissions where you can send in a video of your playing for them to give feedback, for a small fee.
All in all, I think the saxophone is one of the most rewarding instruments to learn. It sits nicely in many genres and will guarantee joy for many years. If you need any recommendations or have questions about picking up the sax, feel free to contact me and you can also check out www.saxo.sg.