Do I need a chest protector or a groin guard? And what can I wear so my upper arm stops getting bruised?
The only extra protection the rules require is a chest protector for female fencers; a groin cup and arm padding are optional. People often confuse a chest protector with the plastron, the half-jacket underarm protector worn beneath the jacket. The plastron is not optional: USA Fencing requires one with every uniform, because it adds a second puncture-resistant layer over the sword-arm side and armpit. A chest protector is a different thing, a rigid plastic plate or cups that take the sting out of hits, worn in addition to the plastron, never instead of it.
Under both USA Fencing and FIE rules, female fencers must wear a chest protector and men may choose to. Foil adds conditions: the protector goes under the plastron, and its outer face must be covered with a specified layer of soft foam carrying an approval mark. For USA Fencing events, older hard protectors can be retrofitted with that foam, but they must then be inspected at equipment check. The foam cover rule is specific to foil, so épée and sabre fencers can use a plain protector. Men who take a lot of chest hits, or are coming back from a chest injury, often find one worth wearing.
No fencing rule requires a groin cup for anyone, and nothing stops you wearing one under your breeches. Hits there are uncommon and nearly always accidental, most often from beginners, from kids whose attacks land low, or from a parry that shoves a blade downward. Plenty of fencers never bother; others, especially coaches who give lessons to children, would not go without one. A combat-sports style cup held by a snug supporter or compression shorts stays put far better than a loose one, though the sideways fencing stance makes some cups uncomfortable. A few fencing brands also sell groin protectors made for the sport.
For bruises on the bicep, elbow crease and inner arm, start by upgrading the underarm protector, since the thinnest budget ones do little to soften a hit and a better one is often the whole fix. Beyond that, a thin padded sleeve worn under the jacket, such as a lacrosse bicep sleeve or a neoprene elbow sleeve, works for many fencers who bruise easily. Keep it slim so it does not restrict movement, and ask your coach before wearing extra padding at a tournament. Fencing kit is designed to stop punctures, not impact, so bruises fade most reliably as your distance and parries improve.