Downtown Hilo is compact and easily walked — the farmers market, the museums, and the old storefronts along Kamehameha Avenue are all within a few blocks of each other, and Liliuokalani Gardens is a pleasant half-hour stroll along the bayfront. For everything else you want a car. Rainbow Falls, the Kaumana lava tubes, Akaka Falls, and the volcano are all short drives in different directions, and none is realistically reachable on foot.
The county's Hele-On bus network connects Hilo with the rest of the island, including Kona and the national park, and fares are low — but services are infrequent and geared to commuters rather than sightseers, so check the timetable carefully before relying on it. Taxis exist and rideshare coverage is thin; for a day without a car, base yourself downtown and treat anything beyond walking distance as an excursion.