At heart, utilitarianism feels like what you get when you ask yourself, "would I rather see few people hurt than many, many people happy rather than few, and how important do I think that to be", answer "I'd rather see few people hurt, rather see many people happy, and this is important", and then apply that systematically. Or if you just imagine yourself as having one miserable or fantastic experience, and then ask yourself what it would be like to have that experience many times over, or whether the impact of that experience is at all diminished just because it happens to many different people. Basically, utilitarianism feels like applied empathy.
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