"Established" really means that your plant has had the time to develop a healthy and sizable root system in its current location. It takes time for plants to become established, and the larger your plants are when they are installed, the more time (and watering help!) they'll need to get there.
During establishment, it's important that plants put more resources into growing their root system. The effect of this is that they will be putting less effort and resources into growing the parts you see above the ground. It's normal for plants to appear to be growing very slowly in the first few years because so much of that growth is happening below the surface. Once they're established, you should see an increase in new leaves and overall size.
The most important thing you can do for your plants to help them establish is to give them plenty of water.
Many of the same rules for watering new plants still apply to watering established plants. You still want to water near the base rather than relying only on sprinklers or overhead watering systems. You still want to give the soil time to partially dry out in between watering. You still want to check your soils and observe your plants for signs of stress.
However, your plants should be capturing more of the water they need from rainfall and should be able to go longer spans without supplemental watering from you. The exact amount of water each of your plants need will vary significantly based on its species, that species' drought tolerance and innate water needs, and your specific site conditions. A plant that gets all-day sun is going to need more water than the same plant that gets afternoon shade, and a plant that sits in a low spot of your yard will likely need quite a bit less water than the same plant on the top of a slope. Soil drainage, humidity, temperature, and time of year will all play a big role. Spend time researching the particular needs of your plants and whether they need more or less water than most.
Your plants, no matter how established, will thrive best with some supplemental water. For certain plants this may only be necessary during very dry spells, while for others you'll get the most out of them with consistent watering throughout the growing season. Proper mulching is also key, as it does a great deal to retain soil moisture and insulate the roots and ground from both extremely high and extremely low temperatures. If you're using drip irrigation as part of your watering plan, you'll want to pay particular attention to grouping plants with similar water needs and to ensure that you aren't overwatering. It's much easier to hand-water a few plants above-and-beyond irrigation than it is to try to dry out others after the fact.
J.R. Thomas Landscaping ~ 10464 St. Rt. 48 Loveland, Ohio 4514 ~ 513-677-4340