How do I rebuild trust and better communicate with the people in my life?

Broken trust is a process to fix.  The biggest things that you have to have are honesty and transparency.  You have to be rigorously honest in your dealings, not "white" lies or lies of omission.  You have to be honest with yourself first and then you can be honest with others.  Honesty with self involves a breakdown in denial.  You have to see yourself as you are and then work on accepting that person. 

Rebuilding trust is very much a process.  You cannot expect trust to be automatically restored just because you are sorry and intend on being different.  The people in your life need to see the changes and that will take time.   Honesty is a way of life. Honesty says I will show up when I say I will show up, that my word means something, and that when I tell you something I plan, I will deliver.   Broken trust makes people that love you skeptical of what you say and do and with good reason.  Transparency says I don't have anything to hide and you can look at my life as close as you want because there are no smoke and mirrors.  Transparency is absolutely essential to rebuild trust in an intimate relationship.  It also gives you the opportunity to show that you have changed and that you are living a life of honesty.  Trust is the foundation of all relationships.  If there is no trust, people do not feel safe and will not allow themselves to be vulnerable with you.

It sounds like you struggle with empathy and communication with others.  Some tips as far as your intimate relationships is to really listen to what the other person is saying, become more attune with their feelings, and provide validation and reassurance in the relationship.  Attunement means you pay attention to the other person and then be responsive to their needs and struggles.  Instead of avoiding your significant other when they are upset, you ask how they are doing and how you can help.  You practice active listening (look it up) and listen more than you talk.  Put yourself in the other person's shoes and try to see things from their perspective.  Be willing to allow yourself to be vulnerable to another person and allow them to see the real you.  It's a movement from self centeredness to seeing others and caring about how they feel and are impacted.  But you have to be less centered on self and more centered on the people you love.  

There are many other issues we often need to address before we can be our healthiest selves, from unaddressed trauma, to anxiety and depression, to negative self image and self talk. Fortunately, all of these issues can be addressed in sustained therapy with a competent therapist.  Better living is possible only when we are at peace with ourselves.