General synonyms:
* Walked slowly and heavily: trudged, lumbered, ambled, shambled, shuffled, marched, toiled, struggled, labored
* Progressed slowly and steadily: crawled, crept, inched, moved slowly, progressed gradually, advanced cautiously
More specific synonyms:
* With a sense of weariness or effort: dragged, stumbled, slumped, faltered, struggled, toiled, labored
* With a sense of determination: persisted, persevered, pressed on, kept going, marched on
* With a sense of monotony or boredom: droned, dragged, ground, meandered, rambled, wandered
The best synonym to use will depend on the specific situation and the nuance you want to convey.