Question itself already tells much. As I see, this is a common issue among new guitarists. My playing more of sounds like hammer drill. tik tik tik... I want to sound more of smoothly changing notes, or somewhere between legato and staccato. What can I do to achieve this? Also, is this a habit to be taken action early on, or as I progress I naturally improve smooth playing?
Edit: Video added as per requests.
When the notes played are on the same string, going higher, don't be in a hurry to play the higher note. The opposite way, again, leave it until the last moment to actually play the lower note, but in this direction, you can usually fret the lower note while the upper one is ringing.
There's also hammer-on and pull-off, which again, work on the same string, thus making it sound very much legato. This should be practised slowly, 60bpm, so one note per second is good for starters. Info. on how to perform both are readily available.
EDIT: one thing I gleaned from the video (thanks) was that you completely take off a finger that will be used very soon again on same string, same fret. Instead, just lift that finger enough to mute the note/string. It's then quicker to re-press. Actually, the playing isn't as staccato as you said in the post! Time and practice will turn it into legato - try playing it all a lot slower, so each note flows into the next. Use of metronome is key here, too.
If the notes are on different strings, chances are for beginner pieces that the next note will be on the adjacent string. Use as much tip rather than pad of fingers, so the notes don't get muted, either way.
Regarding the picking hand, it may be that you're anticipating plucking the next note, so stopping the existing one with the pick, or fingers if that's what you use. So tidy up your picking technique.
All that said - you're already aware that it's a problem, so lots of careful practice will eventually clear up the problem! A video clip of both hands would be of immense help, as I can only answer generally without one.
Quickest tip after seeing the video: The first two notes are the same (F#). The third note is on a different string. There's no reason you have to lift your F# finger from the string as you go to play the new string. As long as it doesn't touch the new string and keep it from vibrating, keep it in place and its resonance will continue. And so on every time you switch strings, when possible. Think of the two notes on the two strings as a "mini-chord": just as you might finger-pick chords with a right hand pattern while holding down the chord with your left hand, in this case you could hold both those frets down while you pick first one string and then the other. You could keep that up all day, back and forth, and have a very sustained tone.
When you play different notes on the same string, there as well you can have more than one finger on the string at the same time. If you have a 5th fret followed by 4th fret, you don't have to wait until the moment you take your pinky off the 5th fret to put your third finger onto the 4th fret. Think about it: the vibrating part of the string is whatever is between the fret you're playing and the bridge. All of the string on the other side of your finger is not vibrating. So you could put other fingers onto the string "behind" your main finger and they wouldn't affect the sound. So in the moment where you play frets 5, 4, and 2 in that order, I would put all three of those fingers onto the string while playing fret 5, and then be able to just take them off one at a time. Part of what is "chopping up" your sound is, if you were to play your video in slow motion, you'd see that between playing fret 5 and 4, there's a few milliseconds in which no finger is on the string. And that means no note. (It's actually more pronounced between 4 and 1.) Actually, just watching your video in slo-mo might make all this more clear to you.
