/ Recommendations for first alpine routes
Aiguille Du Tour, Tete Blanche and Petit Fourche, Pigne D'Arolla are all good and fairly easy without a huge amount of objective danger.
If you don't have those alpine skills, then take a look at the Cicerone guidebooks for ideas of hut to hut treks through the Alps. Favourites include the Walkers' Haute Route (Chamonix to Zermatt) and the Tour of the Vanois.
Hard to answer as you have no profile. We don't know if you are a walker, a scrambler, someone with winter experience or possibly leading VS rock or even E5? My guess is a walker so heading for Arolla is probably a good idea - lots of routes there. Pyrenees are great too.
Good suggestions, though I am not sure I'd recommend the Lagginhorn to beginners when the summit slope is icy. I think jon described descending it as "spooky" in those conditions in one of his posts and I wouldn't want to disagree. Not a place to trip over your crampons.
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