Haste self
There are three related objects that you can use to increase your speed. Speed is a property essential when encountering the more-difficult monsters in melee. The three objects are:
The spellbook grants a level 3 spell of the escape school that increase your speed for 100 to 109 turns.
If you had wounded legs, then the potion of speed will heal them. (Normally, you can heal your legs just by waiting some turns, so do not waste a potion of speed to heal them except in emergency!) Otherwise, an uncursed potion will haste yourself 100 to 109 turns like the spell. A cursed potion will haste yourself only 40 to 49 turns, but a blessed potion will last 160 to 169 turns.
Ultimately though, if you can find speed boots, then the spell and the potion become redundant. A Wizard or other good magic user with intrinsic speed might be able to retain the haste self spell while wearing different boots. Meanwhile, the potion of speed becomes a candidate for dilution to make holy water.
The wand is more useful and important. You can use the wand of speed monster to give permanent intrinsic speed to both yourself and your pets. Hasting your pets allows them to hit more often in melee; your pets will be stronger because melee is their primary activity (other than stealing from shops).
Haste self is also a monster spell; when a monster casts this, they become fast. It produces the message "<monster> is suddenly moving faster."
