A few possible areas:
The European Alps are an area of high mountains across several countries which provides skiing in the winter and good walking in the summer. The whole area is good for wildlife although there are distinctive areas such as the dolomites with their jagged magnesian limestone peaks and specialised flora.As with the Mediterranean islands, plants in the lower regions come into flower first in the spring. Then a wave of flowering moves up the hillside as the snow melts but may not reach the tops until August. Possibly a visit in late June/early July is the best compromise. The lower pastures will be cut for hay and some species not yet out but there should be plenty of high meadows or 'alps' in full flower with gentians and louseworts (Pedicularis sp.) by the tracksides and snowbells (Soldanella sp.) near the melting snow.
There are generally so many wildflowers and other wildlife (including hoopoe, various species of mountain goats etc.) that for a first trip it does not really matter which area is visited. For the more discerning I would definitely recommend 'Mountain Flower Holidays' and 'National parks and reserves of western Europe' [see references at the end] as they list the interesting species likely to be found in each region.
A few possible areas:
Gran Paradiso National Park, Italy [picture above] - It has a particularly rich flora and fauna including species such as Campanula spicata [picture at top of page], alpine ibex, chamois, marmots, golden eagles and eagle owls. We camped near the little village of Pont, there were abundant alpine flowers and insects. There is a route up Gran Paradiso itself [4061m] and several lower level walks.
A little further down the Aosta valley there is the southern approach to the Matterhorn [4482m]. We stayed at Valtournenche which was not particularly pleasant but there were good walks up the valley sides through banks of alpenrose (Rhododendron ferrugineum) with fine views across to the Matterhorn. Summer weather is often hot which means it is best to get up early and do the strenuous walking before the sun is too high.
Further to the east in Austria is the Tyrol. We stayed at Solden [picture right] and walked up some very clearly marked tracks to surrounding villages and huts. This is very much a winter sports area with chair lifts and cable cars criss-crossing the hillsides. Some run over the summer so you don't have to walk up to the high pastures or mountain tops. There were frog orchids (Coeloglossum viride), black vanilla orchids (Nigritella nigra), butterfly orchid (Platanthera sp.), louseworts, gentians, pinks etc. Some of the many butterflies included swallowtails and apollos. There is a variety of habitat in this area with coniferous woodland and rivers, there are also walks for all abilities but I preferred the remoter areas of the Gran Paridiso.
National parks and reserves of western Europe (1982). Duffey, E. Macdonald & Co. London. 288p.
Hamlyn guide to the birds of Britain and Europe (1970). Bruun, B. 319p. 516 bird species illustrated and distribution maps given, a pocket guide.
Collins pocket guide to the alpine flowers of Britain and Europe. (1995). Grey-Wilson, C. & Blamey, M. 352p. 4000 colour illustrations.