I have Herald of Secret Streams, with his ability
Creatures you control with +1/+1 counters on them can't be blocked.
Say I were to put +1/+0 or +0/+1 on a creature, would Herald's ability still apply or does it have to be a +1/+1?
No, it has to be a +1/+1 counter. A counter is a very specific thing, see the comprehensive rules:
121.1a A +X/+Y counter on a creature or on a creature card in a zone other than the battlefield, where X and Y are numbers, adds X to that object’s power and Y to that object’s toughness. Similarly, -X/-Y counters subtract from power and toughness. See rule 613.3.
+1/+0 and +0/+1 counters do not magically combine into +1/+1 counters; the only counters that affect each other are +1/+1 counters and -1/-1 counters:
121.3. If a permanent has both a +1/+1 counter and a -1/-1 counter on it, N +1/+1 and N -1/-1 counters are removed from it as a state-based action, where N is the smaller of the number of +1/+1 and -1/-1 counters on it. See rule 704.
Note that +X/+Y counters other than +1/+1 and -1/-1, like +1/+0 and +0/+1, are very rare; those boosts mostly happen as temporary effects (i.e. a target creature gains +1/+0 until end of turn). This is entirely different from a counter.
The ability refers specifically to "+1/+1 counters", so that is the only thing is would care about. A creature with a +1/+0 counter, or a +0/+1 counter, would not be affected.
It is worth noting that Magic stopped using +1/+0 and +0/+1 counters. Old cards may still have them, but they haven't been seen on any new cards in a long time. Here is an article from 2008 explaining this decision.