For the API or the CLI you could use the flat --dry_run which validates the query instead of running it, like so:

cat ../query.sql | bq query --use_legacy_sql=False --dry_run

Output:

Query successfully validated. Assuming the tables are not modified, running this query will process 9614741466 bytes of data.

For costs, just divide the total bytes by 1024 ^ 4, multiply the result by 5 and then multiply by the Billing Tier you are in and you have the expected cost ($0.043 in this example).

If you already ran the query and want to know how much it processed, you can run:

bq show -j (job_id of your query)

And it'll return Bytes Billed and Billing Tier (looks like you still have to do the math for cost computation).

For WebUI, you can install BQMate and it already estimates costs for you (but you still have to adapt for your Billing Tier).

As a final recommendation, sometimes it's possible to greatly improve performance of analyzes just by optimizing how the query process data (here at our company we had several high computing queries that now process data normally just by using features such as ARRAYS and STRUCTS for instance).

Is there a way I can run the SQL query and export the data to another table/database in GCP, and then run my operations on that exported data?

You can run your SQL queries and therefore export the data into another table/databases in GCP by using the Client Libraries for BigQuery. You can also refer to this documentation about how to export table data using BigQuery.

As for the most efficient way to do it, I will proceed by using both BigQuery and Cloud SQL (for the other table/database) APIs.

The BigQuery documentation has an API example for extracting a BigQuery table to your Cloud Storage Bucket.

Once the data is in Cloud Storage, you can use the Cloud SQL Admin API to import the data into your desired database/table. I attached documentation regarding the best practices on how to import/export data within Cloud SQL.

Once the data is exported you can delete the residual files from your Cloud Storage Bucket, using the console, or interacting with the Cloud Storage. API

Am I correct in assuming (and I could be wrong here) that once I export data to standard SQL DB in GCP, the cost per query will be less in that exported database than it is in BigQuery?

As for the prices, you will find here how to estimate storage and query costs within BigQuery. As for other databases like Cloud SQL, here you will find more information about the Cloud SQL pricing.

Nonetheless, as Maxim point out, you can refer to both the best practices within BigQuery in order to maximize efficiency and therefore minimizing cost, and also the best practices for using Cloud SQL.

Both can greatly help you minimize cost and be more efficient in your queries or imports.

I hope this helps.

Assuming you are using default on-demand pricing plan, the cost of a query is $5 per scanned TB in US region as seen in BigQuery pricing page.

You then get the cost of each job with:

select 
  job_id, 
  ifnull(5. * (total_bytes_billed / pow(2, 40)), 0) as job_cost
from `region-us`.INFORMATION_SCHEMA.JOBS_BY_PROJECT

(1 TB = 2^40 bytes)