Performing some preliminary calculations for an aircraft design I have and was wondering what thrust I should use to calculate thrust to weight? The thrust calculations I have, vary due to speed (since in general force is a change in linear momentum) and so the faster the aircraft is traveling, the lower the thrust I have is. Should I use the initial thrust when the aircraft is initially accelerated, an average thrust of some sort or a different thrust estimate altogether?
Well, it depends on what you want to know. If you are interested in thrust-to-weight ratio in certain conditions, you should use thrust (and weight) in those conditions.
In terms of standard performance specifications, the maximum static thrust (zero speed, zero altitude, ISA conditions) is normally used. This is often the only thrust point specified anyway. The weight is often less certain; it would be fair to use MTOW for this figure, but some "practical" weight/configuration is often used instead, because it gives better (and possibly "more relevant") result. Technically, the conditions should be explicitly specified.
If you are interested which thrust formula is more correct, this is a separate question.