You are touching upon many different issues here. Many are deep enough subjects that one could write a book on each topic.
Here are some thoughts:
1. Yes, get a better tuner. Each tuner type has its benefits and shortfalls. Good idea to learn when to use what kind (for example, clip on vs. for example a Boss TU12-H, etc.). I only use a clip on (Snark - the red one) for quick tuning, etc (not for recording). if I'm not going to use my ear. For more serious tuning, I use the Boss. MANY people use this style of boss tuner and some use things like clip-ons for live.
2. Yes, learn to tune very well by ear. Critical.
3. Understand the difference between tuning your instrument, intonation, and the limits of each.
4. Understand the difference between the different ways of tuning your instrument by ear (harmonics, open strings, octaves, etc.).
Many things impact tuning an instrument including you and the instrument. Playing out of tune is a terrible thing and we all have done it and many famous recordings are out of tune. But, we all wish we could improve it. So we try.
I went through a long period of frustration and struggle on this topic. I would hate to have to see someone else go through it considering how much info there is now available online. A deep topic, but VERY interesting and will help you immeasurably as a musician in learning, live, recording, practicing, music history, everything.
Just my 2 cents (no pun intended - maybe you get it or will soon enough???).
LL