Short answer? Yes.
Can we better love others by loving God? Yes. There is no contradiction in the verses you presented.
John 13:34 A new command I give you: Love one another. As I have loved
you, so you must love one another.
What's the relationship between loving God with all my heart, soul and might and loving my neighbour as myself?
The relationship is explained in the teaching that one will follow the other and is taught to new Catholics in RCIA. (I spent about 6 years serving in that ministry). A useful visual image to hold in your mind's eye as you consider this is that of the Cross. It has two elements: the vertical piece, which represents our personal relationship with God (Commandments 1-3) and the horizontal piece, which represents our relationships with our family, friends, and neighbors (Commandments 4-10). They are joined in our heart. (Ref: Teaching point from our RCIA ministry)
CCC 2081 The Ten Commandments, in their fundamental content, state grave obligations. However, obedience to these precepts also implies obligations in matter which is, in itself, light.
2082 What God commands he makes possible by his grace.
In abbreviated form, His commands to obey Him are made possible by His grace (He won't command you to do what you can't do; see Philippians 4:17) and any burden upon the faithful is light, or easily born.
The Catechism of the Catholic Church organizes the Ten Commandments (Section II of IV in the Catechism) into two groupings, headed under the following: Commandments 1-3 are presented under the heading
CHAPTER ONE YOU SHALL LOVE THE LORD YOUR GOD WITH ALL YOUR HEART, AND
WITH ALL YOUR SOUL, AND WITH ALL YOUR MIND
While Commandments 4-10 are under the heading
CHAPTER TWO YOU SHALL LOVE YOUR NEIGHBOR AS YOURSELF
The Catholic Church teaches explicitly that what Jesus said about the two greatest commandments is an important truth: All the Law and the Prophets hang on these two commandments. That means that if you apply yourself in adhering to His commandments (also called the Decalogue) the rest will follow. His Grace is enough. (2 Corinthians 12:9)
To summarize pages upon pages of what's at the link covering all ten commandments in the Catechism:
- When one loves God one will obey him (without it being a burden, CCC 2081),
- Jesus commanded his disciples to to love one another as I have loved you (per the verse cited in John)
- Jesus is one of three persons in the Trinity, so his command is from God, and is to be obeyed
- The form of love being referred to here is agape (divine / sacrificial love)
- God provides us with the Grace necessary to obey his commandments - through loving Him, your heart will be open to love your neighbors.
That's a very brief treatment of pages and pages of Catholic teaching. I'd recommend reading the entire Section II of the Catechism for more detailed treatment of how obeying God opens up the door to being a loving, selfless Christian.