The word in Spanish for empathy is empatía, and the word for sympathy is simpatía. You can combine either noun with verbs like tener (to have), mostrar (to show), or expresar (to express), among others:

La gente le tendría simpatía y admiración al mismo tiempo. Y hasta lástima.

People would feel sympathy and admiration for you at the same time. And even pity.

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To use the verbs mostrar and expresar instead of tener, you might say:
 

La gente le mostraría simpatía | People would show sympathy for you.

La gente le expresaría simpatía | People would express sympathy to you.

But how can you directly express your sympathy to a person? The expressions te tengo simpatía ("I sympathize with you" but also "I like you" in some contexts) and soy empático contigo (I'm empathetic toward you) are correct but not very colloquial. You can use other expressions instead, for example, estoy contigo (I'm with you):
 
¿Confías en mí? -Sí. Yo estoy contigo.
Do you trust me? -Yes. I'm with you.

Another good way to show support is by simply saying te apoyo (I support you):

Ay, amigui, yo te apoyo.

Oh, friend, I support you.

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In the case of more serious situations, for example, when receiving bad news about something, the most common way to show your support is by saying lo siento mucho (I'm very sorry), or the more emphatic cuánto lo siento (literally "how sorry I am"). There are different ways to use these phrases, depending on what you want to say. For example:
 
Mi papá está muy enfermo. -Oh, lo siento mucho.
My dad is very sick. -Oh, I'm so sorry.
 
Siento mucho que no puedas visitar a tu familia ahora.
I'm very sorry that you can't visit your family right now.
 
¡Cuánto lo siento que tuvieras que pasar por eso tú sola!
I'm so, so sorry that you had to go through that all by yourself.

Just pay close attention to the context and tone because, like in English, lo siento is also used commonly in less serious situations:

Lo siento, pequeña, pero aquí las cosas hay que ganárselas.

I'm sorry, little one, but here things have to be earned.

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You can also use lo siento mucho to offer your condolences. Altenatives include te ofrezco mis condolencias (I offer you my condolences) or recibe mis condolencias (receive my condolences), expressions that many people shorten to just mis condolencias (my condolences), or mis más sentidas condolencias (my heartfelt condolences):
 
Mis condolencias, Sr. Gutiérrez. -Gracias. 
My condolences, Mr. Gutierrez. -Thank you.
 

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Finally, showing support is also about extending a helping hand, right? In Spanish you can use expressions such as ¿en qué te puedo ayudar? (how can I help you?), ¿te puedo ayudar en algo? (can I help you with something?), cuenta conmigo (you can count on me), estoy para lo que necesites (I'm here for whatever you need), among others. A very colloquial expression is echar una mano (to lend a hand):

...para echarle una mano a la familia.

...to lend a hand to the family.

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