What plants/grasses can I plant to entice the ladybugs in my garden to stay? They are great for controlling aphids, but seem to move on pretty quickly.
It's going to be difficult to keep them in your garden over the breeding season if there isn't a constant food source. However, if you find out what species of ladybug you would be likely to find in your area, you should be able to learn if it has a preferred habitat type or host plant. For example, the UK ladybird survey has posted this useful information: http://www.ladybird-survey.org/habitat.aspx
If ladybugs overwinter in your climate, you can encourage them to stay in your garden by providing shelter over the cold season. Last winter I left small piles of sticks, bark, rocks and leaves at a few sheltered areas in my back garden and found many ladybugs emerging in the warm spring sun. I've found that they especially like hunkering down in dried, hollow stems of dill and fennel, so if you grow them, or have some hollow bamboo canes, add them to your ladybug winter shelter.
Plants ladybirds like looks promising:
According to this site, ladybugs have trouble obtaining nectar from many plants because they have small mouths. It mentions that the flowers on the herbs dill, angelica, cilantro, fennel, celery, caraway and tansy are all easily accessible to them.
If you are willing to avoid killing your weeds, the site also mentions that wild carrot, dandelion, lambsquaters and clover have flowers that ladybugs can obtain nectar from.
For vegetables, the site mentions parsnips, beans, peas, and cabbage, and notes that some ladybugs will even eat these plants if food is scarce.
Finally, "spirea, buckwheat, cosmos, coreopsis, scented geraniums and yarrow are other types of flowering plants that ladybugs are drawn to." It also mentioned that white cosmos is especially attractive to them.