Compression means more rows stored on a page, reducing I/O by reducing the number of pages needed to retrieve a value and the number of pages needed in memory.
Columnstore is a completely different storage mechanism (although it also has compression) that is reorienting your data storage from rows to columns. This more directly supports analytical style queries (GROUP BY, SUM, AVG, etc.) while sacrificing performance of point lookup style queries (one row or one small set of rows based on tight filtering criteria).
These are not interchangeable parts, compression & columnstore. They do different things. They have different purposes. Could one help you more than the other? Yes. However, you have to determine what the problem is in the first place in order to understand which of these mechanisms is more likely to help. For example, if you primarily have point lookup queries, implementing columnstore won't help performance and will likely seriously, negatively, impact that performance. That assumes you use a clustered columnstore. If you just add a nonclustered columnstore and you're running point lookup queries, chances are, you've sacrificed a lot of disk space and added a bunch of overhead to loading the data, but the optimizer will just ignore the index in favor of those that better support point lookups.
I wish there were easy answers when we start talking about performance tuning. If A do Z. If B do Y. However, it's just rarely that simple. You have to understand what the purpose of your queries is. You have to understand the structures that you have in support of those queries. You have to understand how SQL Server works to use the structures in support of those queries. Then, you have to understand all the options & methods that can better support your queries and structures. There isn't a shortcut like put NOLOCK on every query, enable compression, or switch all storage to columnstore. Any of these, situationally could help. Equally, any of these, situationally, could be neutral or could hurt. We have to understand the situation in order to better apply a solution.