I'm assuming that by "each month" you mean a calendar month.
One way:
select id, logfile, min(timestamp)
from MyTable
group by id, strftime('%Y-%m', timestamp);
This relies on SQLite-specific guarantee whereby, when min() or max() aggregate functions are used, other selected columns are taken from the same row on which the minimal or maximal value is found.
Another way that doesn't rely on this property, but would be much slower on a large dataset:
select * from MyTable t1
where not exists (
select 1 from MyTable t2
where t2.id = t1.id and t2.timestamp < t1.timestamp
and strftime('%Y-%m', t2.timestamp) = strftime('%Y-%m', t1.timestamp)
);