Red and barely visible splits two ways. If the plant is suffering, the usual culprit is a spider mite feeding on it. If the plant is fine and the red things are on the wall, the path or the window frame, they are not feeding on that plant at all. Hold a sheet of white paper under a suspect leaf and tap it: spider mites shaken onto paper start moving about quickly, and a hand lens shows you what you have.
The rest of what gets called “tiny red bugs on plants”. Some are pests we have not profiled yet; some are insects doing you a favour; some are not insects at all, and no treatment will help.
Reddish, brown or dark green, about 0.75 mm, with a very long front pair of legs — and the houseplants are untouched. They feed on lawn grass and clover, and in autumn and spring come in through cracks around windows, especially on sunny south-facing walls. They do not bite and do not breed indoors, where they die within days. Crushed, they leave rusty red stains, so vacuum rather than squash.
Velvety red mites of the soil and leaf litter (*Trombidium* and relatives). The adults and older nymphs are free-living predators; the larvae ride on insects such as aphids, and on spiders, as parasites. Nothing on your plants needs protecting from them, so leave them be.
Round and translucent, like tiny droplets, turning cream before they hatch, and not moving. They sit on the leaf underside with the mites. Only some miticides kill eggs, and UC IPM notes that most miticides must be applied more than once to give control.
Two-spotted spider mite →Adults are about 1/2 inch long, black with red or orange markings. The newly hatched nymphs are tiny (about 1/16 inch) and bright red, turning red and black as they grow. They feed mainly on boxelder, maple and ash seeds without noticeable harm, gather on sunny south- and west-facing walls as it cools, and are a nuisance indoors rather than a pest of plants.
Boxelder bugs →If itchy bites around the ankles or waistband came first and nothing was ever seen on the plant, think chiggers. The biting stage is a larval mite about 1/150 inch across, orange, yellow or straw-coloured, that waits on vegetation in damp spots with tall grass and weeds. The adults are brilliant red but not parasitic, and are often seen on lawns and pavement. A lawn-and-ankles problem, not a houseplant one.
Four questions about the damage separate these better than an appearance can, and this page answers some of them for you.