Recently I was asked how a new Shopify store should get started with their marketing. They were considering SEO and Google Ads but ran into problems with both and wasn't sure where to start.
Roughly I'd recommend getting your analytics, structured data, and Rich Results started first and then then look into SEO or Merchant Center (for paid or free ads).
1st: Add your Google Analytics to your Shopify settings and turn on Enhanced Ecommerce tracking. That'll handle 80% of what you need on the analytics side.
Once you have a supply of regular orders every day, it would make sense to look into Repeat Customer Insights to analyze your customer behavior.
2nd: Get your structured data working with JSON-LD for SEO so you can get Rich Results. This isn't as important early on but the process is mostly automatic and it can take weeks or months before Google awards Rich Results. It's a process you want to do early and work on other things while Google chews on the data.
3rd: Go through Search Console and fix any errors it says. Review warnings and recommendations it gives to see if you want to implement them but be careful about structured data warnings as Search Console has bugs around those and will recommend things that'll hurt your SEO.
4th: Decide if you want to tackle paid ads (Merchant Center) or go with a lower-cost strategy like SEO. Both paths can work but it'll depend on your finances.
SEO-path. Go through my Shopify SEO guide and verify and fix everything included in it. That includes coming up with a link building plan and executing on it over the next few months.
Ad-path. You'll want to setup and test your Merchant Center data feed but since you have solid structured data now (2nd step) that shouldn't be a major problem. You might need to add more data into Shopify as Merchant Center needs a lot of product data that many stores don't setup. You might also need to strip out any bad data that your theme supplies or any Google violations from its code.
Look at one issue at a time and try to understand what Merchant Center needs, make a fix, and then give it a week or two to notice the fix. Rushing a lot of changes at once can make it difficult to know what caused new problems.
Once you have those done, you should have a solid marketing foundation. There are other things you can work on as a continuous-improvement basis too. I go into the SEO ones in the later sections of my Shopify SEO guide while a Build/Test/Review loop works wonders for optimizing your ad campaigns.
This stuff will take you a few months to get going so don't stress if progress seems slow. SEO moves very slowly but it accumulates over time.
Analyze your customer behavior
A quick and automatic way to measure your customer behavior is with Repeat Customer Insights. It'll automatically import your Shopify data and run a number of analyses to highlight different customer behavior.