By bus
Bus is by far the cheapest option but could involve many hours of travel depending on your starting-point. As a guide it takes roughly six and a half hours hours by bus from Jinan,capital of Shandong to the bus station in Weihai. There are many companies serving Weihai with many destinations possible at good frequencies.
Air
The airport is approx 45 KM to the City Centre. The rather new toll highway somewhat reduces the travel time to or from the city centre yet adds slightly to your overall fare. Via tollway or not, expect to pay about 100 - 130 yuan each way.
A backup option is nearby Yantai with a taxi into Weihai. However, the journey can take an hour, and the taxi fare could be about double, sometimes involving two taxis depending on the driver. The first taxi may take you only as far as the beginning of the toll road where you'll easily find a second taxi to your destination. The rationale is the Yantai driver is unlikely to find a customer for the return.
Domestic flight services are from Beijing, Shanghai both Pudong and Hongqiao and other large Chinese cities. There are more frequencies and origins to Yantai than to Weihai which particularly experiences reduced schedules during the winter months.
Another option is Incheon to Weihai or Yentai. It's a one-hour hop but has the amenities of an international flight. Attempt timing your international arrival into Incheon with the least amount of layover. Transit through Incheon is a breeze. Korea is 1 hour ahead of China, so keep timezones in mind when organizing your flights. The flights to Weihai depart about 9am, and noon for Yantai. Asiana and Korean Air serve Weihai although China Eastern code shares with KE.
Incheon is a good option if schedules align as this routing may sometimes yield significant savings even with an overnight stay. There is not much to see near Incheon, but usually a shower and a bed are about the only wants after long haul travel anyway. From the US, late night flights from Los Angeles, Chicago and New York to Incheon run on certain days which coordinate nicely with the onward 9am Weihai departure.
If you overnight in Incheon, there is a nice Hyatt among other hotels just beyond the airport parking lot. The hotels have free shuttles about every 20 minutes at the customer exit # 14, the furthest exit to your right as you clear international arrivals. The Hyatt would be walking distance if not for luggage or jet lag. Your luggage may be checked on through to China and not offloaded. Check with your carrier's policy and pack a carry-on, as appropriate. Luggage checked through to destination removes baggage claim and simplifies customs in Incheon.