Damage from the adults is generally not severe. The larvae are the little buggers that wreak havoc, they feed voraciously on roots and can kill or weaken many plants. Increased susceptibility from their damage to root diseases such as Phytophthora is very common.
Remove mulch and other plant debris to eliminate hiding places from around plants. If they are a voracious problem cut down on watering, water only when necessary as they prefer moist damp conditions. Hand removal of weevils is somewhat effective in small plantings or container gardens. Remember they are nocturnal and won't be seen much in daylight.
Parasitic nematodes a\k\a beneficial nematodes are especially effective against the weevil larvae. The nematodes will help curtail their population expansion. For successful use of nematodes you should allow sufficient time for multiplication of the nematodes in hosts (weevil larvae) and dispersal of nematodes throughout the soil. Early- to mid-May application is optimal. The larvae are also attacked by some general predator insects including Rove and ground beetles such as carabid beetles.
Barriers such as Tanglefoot and diatomaceous earth are helpful in fighting off weevil infestations , as well as against many other garden pests. Tanglefoot, or similar products applied to trunks will prevent these flightless beetles from feasting on foliage.
For ornamental plants place burlap, or even plywood around the tree and shrub bases to trap weevils that hide under it during the day. Remove the burlap periodically and slaughter the little buggers. Another trap is known as a pit fall trap, which is easily made by placing a paper or plastic cup in the ground, leaving the rim at soil level. Bend a piece of paper to a cone shape near the top and insects will slide down into the cup filled with water and drown.
Milky Spore is another weapon in the war against weevils , it is not harmful to beneficial insects, birds, bees, pets or man. The product is approved and registered with the EPA, Milky Spore will not affect wells, ponds or streams. As larvae and grubs burrow through the soil, feeding -they ingest the spores . The spores germinate inside the grubs, and multiply rapidly in their blood. When the bacteria become very numerous in the blood they again form spores, completing the bacterial cycle. Infected grubs survive for 7-21 after their blood has become loaded with spores, particularly in low temperatures. As the grub carcass decomposes, it releases billions of new spores. It is only effective against newly hatched larvae within the soil, not once they enter the plant.
Bacillius thuringiensis [Bt] is somewhat effective against the larvae stages it should be applied as soon as the presence of adults is evident, to head off the imminent egg laying and hatched larvae.
Neem based Products, such as Azatin or Azadirachtin [extract of the neem seed-] prevents the larvae from developing normally and is also a good alternative for later larval stages. These products degenerate rapidly in nature and need to be reapplied frequently. For effective use mix Neem oil with equal parts of water and saturate the ground around the plants, allowing it to soak in thoroughly.
In severe infestations, there are a number of broad spectrum pesticides registered for use against weevil, sold under varying trade names. Acephate is one I would recommend. Bonide 941 and 951, Orthene 97.4% Acephate and Agrisel are a few brands readily available to home gardeners.