I have a beginners question. Best method for building up Instagram and traffic to a pet site.

Since you are looking to build your site organically, you may want to try the Gary Vee Instagram challenge. You do have to apply what he is talking about and give it time to grow your site.

The #1 thing you should do with articles is share them on social media. Take the time to set-up as many appropriate accounts as possible. The wider you spread the word the more traction you will get.

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Instagram - search for people relating to the pet topic and follow these people. They will likely see that you've followed them and some will follow you back. Then you can advertise your articles by posting on Instagram and some of your followers will click through.
In terms of your articles, if you want them to rank and show in organic search results then you need to set each article up with heading, meta description and content to include the keyword you're targeting. Include good infographics and images and a FAQ section at the bottom of each article and slowly you should see organic traffic and each articles rank improve.

I would set it up like this based on what you are trying to do.

1. Create a blog where you post your articles.
2. Create a Facebook page/Instagram account that posts smaller, shareable, bite-sized versions of those articles that link to the blog post. Keep it simple, go to Canva and pick a theme that works with the brand and just write simple text on a few slides and then a little description in the body of the Instagram/Facebook post and add the necessary hashtags.
3. Use the methods discussed above
4. For more content, take the 10x10 approach. Pick ten topics in your niche, write out ten different points about each topic and make a post about each point. That will give you 100 pieces of content easily.
5. To supercharge this, run a "page like" campaign on a small budget of about $3-5 dollars a day.

A note, these shareable posts should just be about giving information. And AT MOST, a "soft sell" within your articles.

There are services you can buy, contractors, and whatnot. But that is not cheap for what you are trying to do. A simple FOLLOW?UNFOLLOW campaign may do you good.

Or my personal favorite. Find your competitors, find who likes/follows them and send them a message or reply to their comments. This is an aspect of what Gary Vee was talking about. People will naturally look you up this way, and you will start to gain traction with enough time invested and value given.

Good luck!