Is it normal for my bp to shoot up to 170ish / 110ish or higher along with dizziness during only mild exercise like push-ups / dumbells? concerns me.

Normal vs. Healthy?: Big difference. Normal only means usual, common. Healthy=a very different issue. Basic issue: any BP reading from only a single heart (Systolic) & another single beat (Diastolic) within limits of accuracy of measurement system being used; digital meters least accurate (literally guess). Situation, as described, too vague for good answer. Exercise tends to result in ↑SBP, varies with exercise, etc.