One of the guidebooks that described this trail made this a very confusing hike and I would have been better off looking more at the map than the trail description. Let me see if I can make this less confusing.
This mountain is famous in Alaska because every 4th of July, since 1915, there has been a race up the mountain. The official trail for this race is oddly called Racer's Trail. It is very steep and demanding and the first part is the hardest as you have to climb up a rocky cliff.
With me so far? There is also an easier hiker's trail that starts at the base of the mountain, switchbacks up the mountain a ways on an old road then goes sharply right, follows a ridge line and then heads up the gently sloping spine of the mountain.
You can access the racer's trail, and avoid the first hard part, on the hiker's trail before it turns right.
I wasn't able to do the hiker's trail as it still had too much snow (too many trees block the sun on the ridge so the racer's trail melts faster). The racer's trail is very unforgiving but you do get great views of Seward, Resurrection Bay, the rocky islands, and the Kenai Mountains. Once you get above treeline, the view really opens up but the trail gets harder as you must grapple with rocks and extremely steep sections.
The "easier" hiker's trail (also known as the Sheffler Creek Trail) starts at the corner of 1st and Munroe. The racer's trail can be accessed off the end of Jefferson Street.
If you do the racer's trail, it really is steep. Some parts of the trail are worn and smooth from all those runners and can be quite slippery. There are sections of the last part that I had to use my hands as well. I would definitely bring a walking stick.
Just up the hiker's trail is a waterfall. Do not fork here. A little further up (head left), you will see the sign (right-falls/hiker left-racer).
Bring a wind breaker and fleece. It can get really cold up there and Seward can see blue skies quickly turn to rain.
In the Good Friday Earthquake of 1964, half the town of Seward was wiped away by the giant tsunami caused by the quake's massive force.
I try and try but I will never understand people who get enjoyment out of running up mountains.
Two things in Seward you may want to consider before or after this trip; sea kayaking with a guide and/or going on a boat tour. I highly recommend either. There are many reputable outfits and tour groups with 3 hour to 3 day packages. Link to the Seward Chamber of Commerce for more information.