Is it generally faster to select into a temp table than selecting into an actual table?
writing to tempdb is faster than an actual table not in tempdb


It's true.  There are two IO enhancements in TempDb.  

Writes to a table in a user database must have their log records flushed to disk on commit, or if a minimally-logged insert (like SELECT ... INTO), must have the database pages flushed to disk on commit.  The way the minimal logging works in a user database is that the database pages are written directly to disk.  By the time the SELECT ... INTO completes, the new pages have all been written to the physical files.

TempDb writes don't have to be flushed to disk on commit since TempDb is never recovered after a failure.  So they simply aren't.  Your changes generate log records, but the log buffer is flushed to disk only when it's full, not for every commit.  

And since SQL Server 2014 the minimally-logged inserts in TempDb aren't always written to disk either.  If you load a small, short-lived temp table it may never be written to disk at all.  The log will have a few records about the page allocations and metadata entries for the table, but that's it.

EG run the following batch in tempdb, a full recovery database and a simple recovery database to see the differences.

drop table if exists foo
go
declare @data bigint
declare @log bigint 
select @log = sum(case when type_desc = 'LOG' then num_of_bytes_written end) 
      ,@data =  sum(case when type_desc = 'ROWS' then num_of_bytes_written end) 
from sys.database_files f
cross apply sys.dm_io_virtual_file_stats(db_id(),f.file_id) fs

select * 
into foo
from sys.objects 

select -@log + sum(case when type_desc = 'LOG' then num_of_bytes_written end)  log_bytes
      ,-@data +  sum(case when type_desc = 'ROWS' then num_of_bytes_written end) data_bytes
      , (select recovery_model_desc from sys.databases where database_id = db_id()) recovery_model
from sys.database_files f
cross apply sys.dm_io_virtual_file_stats(db_id(),f.file_id) fs


and you'll see something like:

For simple recovery:

log_bytes            data_bytes           recovery_model
-------------------- -------------------- ---------------
24576                16384                SIMPLE


for full recovery:

log_bytes            data_bytes           recovery_model
-------------------- -------------------- ---------------
36864                0                    FULL


and for tempdb:

log_bytes            data_bytes           recovery_model
-------------------- -------------------- ---------------
0                    0                    SIMPLE


Sometimes for tempdb you'll see the log buffer flushed:

log_bytes            data_bytes           recovery_model
-------------------- -------------------- ---------------
61440                0                    SIMPLE