If you had to choose one metric to watch for measuring new customer performance, I'd pick profit from new customers.
This is based on a core business metric (profit) but segmented for new customers only. It would include the costs to acquire the new customers (CAC) and any overhead costs too.
As long as it's positive, trending up, and a good enough return, you'll know your new customer process is working well.
If costs are difficult to separate and calculate for new customers, alternatively revenue from new customers could fill in. You'll want to keep track of your total expenses too and use that as a break-even target (e.g. $200,000/mo in expenses means revenue should be at least $200,000 each month).
Actual profit is better though if you can get to it easily.
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.
Market to your customer's timing
Figure out how long customers wait in-between purchases and you have a key component for your marketing timing. This is the basis of the Average Latency metric and Order Sequence Report in Repeat Customer Insights.