The shape and surface of roots
Two architectures
Taproots are a dominant single root driving deep with smaller side branches. Carrots and dandelions are taproot plants. A taproot anchors the plant against wind and pushes deep into the soil. Fibrous roots form a dense web of many small roots near the surface, with no single dominant root, as in lawn grasses. They stay shallow and spread wide.

Fig. 3. Two ways to build a root system. The taproot on the left commits to one dominant root and drives it down; the grass on the right spreads a dense web of many small roots near the surface.
Where roots actually drink
At the growing tip: a root cap protects the point as it pushes through substrate (and senses gravity, so roots know to grow down); a zone of elongation where the root lengthens; and a zone of differentiation where cells specialize and root hairs emerge.
Root hairs are single cells growing out from the root's surface that widen its contact with the substrate, though experiments disagree on whether they add much to water uptake (Duddek et al., 2023). In most plants each one lives only days to weeks, and new ones keep forming as the root grows on through the substrate (Moore & Bradley, 2022). Root hair lifespan isn't fixed: in two cotton cultivars, median lifespans ran from about 15 to 27 days depending on the cultivar and how much phosphorus the soil held (Zhang et al., 2021). That's why we handle the rootball gently when repotting: we'd expect rough handling to strip off root hairs and fine roots that the plant then has to regrow. That's our reasoning, not a measured result.

Fig. 4. The root tip. Root hairs emerge in the zone of differentiation, just above the zone of elongation.
A footnote on partners
The great majority of flowering plants, roughly 85 percent of species, form root partnerships with soil fungi, called mycorrhizae (Brundrett & Tedersoo, 2018). Depending on the kind, the fungus either wraps the root in a sheath of fine threads or grows into the root's cells (Dunn et al., 2017). In the most common kind, arbuscular mycorrhizae, the plant feeds the fungus carbon; in exchange the fungus's fine threads, its hyphae, gather mineral nutrients out in the soil and carry them back to the root, sometimes over many centimeters, phosphorus above all (Smith & Smith, 2011). Phosphorus moves through soil very slowly, so that reach is a real advantage. Two things to know as a container grower: most soilless potting mixes contain no mycorrhizal fungi, so a container is where adding them could make sense, and some plants, cabbage, kale, broccoli and spinach among them, show no response to mycorrhizae at all (Dunn et al., 2017).
Strengthen your understanding
Try these in your head first, then open to see.