Depends on the kind of romance you want to write, as well as the nature of the relationship and the personalities and behaviors of the characters in question. There are so many different ways to approach writing a romantic subplot, but, knowing nothing about your story and characters, here are my two cents.
Love is shown not only through words or physical reactions like a racing heart or blushing. Actually, those could be also attributed to infatuation or physical attraction rather than love (granted, romantic love can start with infatuation, and accompanies physical attraction, but if you want to be subtle you probably won't emphasize these a ton at the beginning of the relationship). If we define love as "genuinely caring for the other person's wellbeing and placing it above one's own", you could show your couple's budding romance through their actions and behavior towards each other accordingly.
If you love someone, you'll go out of your way to help them. You're invested in them, their personal life, their interests, their problems. Their struggles become your struggles, because love at its best is a kind of close partnership, where you share yourself completely with the other person and vice versa.
So you could develop the romance by having your characters act this way together. If it's a subtle development, they'd probably be friends to begin with and grow closer over time. They make themselves available to the other, and go out of their way to help each other with their problems. They support, encourage, and comfort each other, feel at home with each other, and open up about secrets and personal things that they wouldn't feel comfortable sharing with anyone else.
A lot of this is characteristic of a good friendship, but when they start to become "more than friends", they might become more physically affectionate and possessive, wanting to spend time alone together, and feeling uncomfortable if other people get too close to their partner. People with hidden feelings might find excuses to spend time with or touch the person they like, for example, hugging them for a little longer than normal, holding their hand when they don't have to, brushing against them when they walk together.
Lots of ways to hint at unacknowledged feelings without saying "my heart thumps". :]