Does having an accent matter when you speak Spanish?

Having an accent doesn't matter much for being understood in Spanish. What matters is being clear. If your vowels are clean, you put stress in the right place and you speak at a steady rhythm, people will understand you with a strong English accent. Sounding native is a bonus, not a requirement.

Accent vs. clarity

These are two separate things. An accent is how your pronunciation shows where you're from. Clarity is whether the listener can figure out the words. You can have a heavy accent and be perfectly clear, or a light accent and be hard to follow.

I think learners spend too much energy trying to lose their accent when they should aim for being easy to understand. Native speakers have accents too, and Spanish has many.

What to get right

If you only fix a few things, these give the most improvement:

Pure vowels. Spanish has five short, steady vowels. English slides them. Fixing this changes your sound more than anything. See the hardest sounds.
Word stress. Hablo (I speak) versus habló (he spoke) differ only by stress. Putting the emphasis in the wrong spot can change the meaning, or make a word hard to recognize. Accent marks tell you where it goes, which is partly why accent marks matter.
The tap and trill r. Pero and perro are different words. See rolling your R.
Rhythm. Spanish is more syllable-timed, with each syllable a similar length. English squashes unstressed syllables. Even speech sounds more Spanish.

What matters less

Don't stress about:

A perfect rolled r on day one
Exactly copying one country's accent
The small differences between regions
Never sounding like an English speaker

Most people will be kind about this. Many native speakers find a learner's accent charming, and they care more that you're trying.

Is there a "right" accent?

There isn't just one. Spanish spoken in Mexico, Spain, Argentina and Colombia all sound different, and all are correct. Pick the one you hear most or that you like, and don't worry if you blend a bit. There's more on this in mixing dialects.

When it can matter

Accent can matter if you're aiming for a job where clear speech is key, like teaching or presenting. Or if you simply care about it, which is a fine personal goal. Then it's worth training with recordings and feedback. But for everyday use, travel and friendships, you're fine.

How to improve it anyway

Listen a lot. Your ear leads your mouth. I'm biased since I made Langtok, but slowing a short story to 0.75x while the words highlight as they're spoken is a good way to train it.
Shadow short clips to copy rhythm and melody. See shadowing.
Record yourself and compare with a native speaker saying the same phrase.
Ask for feedback from a tutor or friend: "Which sound is hardest to understand?"

Accent improves slowly in the background as you listen and speak more. The best plan is to work on clarity and let the rest come with time.