What would be the best way to model my simple table? What would be faster: searching for the right row through a table with 17 different columns, each with a tinyint value of 1-4?

Or, searching for the right row through a table with only 1 column with those 17 numbers concatenated together as a single bigint?

The numbers when combined are all unique from one another (no row has the same values/value as any other). I'm new to indexes and what not, so I have no idea how I should model my table/server logic.
If we're talking about something where you may want to query on one or more of the attributes my suggestion would be multiple columns, indexed appropriately (for a small, infrequently-changing data set, index all the columns).
This works best for the case where you may want to run queries like
SELECT * FROM table WHERE attribute1=1 AND attribute3=2
since you will have several indexed columns and searches on one or more of them will be relatively efficient.

If we're talking about something where you only ever want to query on the aggregate (all attributes match) a single column will be more efficient in terms of storage, and likely in terms of indexing as well.
This works best for cases where the previous example would always devolve into
SELECT * FROM table WHERE attribute1=1 AND … AND attribute17=17
as rather than having to cross-check across multiple columns it will simply be
SELECT * FROM table WHERE attributeColumn=1234567  -- Rather than having to scan 17 indexes (or examine the contents of 17 columns in every row) you will just have the single index scan.