Want to see Alps mountain from nearby areas

The Alps are very large, so it depends on where and how you want to see them.

IF you want to this "quick and easy" you can go to Germany's tallest mountain which is the Zugspitze right on the Austrian border. You can go all the way up with a cable car. Live cam at https://zugspitze.de/en/News/Live-Info/Zugspitze#

It's actually very easy, if not always really cheap. You can get very close to breathtaking mountains by public transportation in France, Switzerland, Austria, or Italy… Towns that are reachable by train or bus and are in the Alps include Chamonix in France (connection from Saint-Gervais), Zermatt in Switzerland. Grenoble in France or Innsbruck in Austria are larger cities that are also very close to mountains.

If the weather is clear, you will have beautiful views straight from the center of the town and multiple easy walks or cable car journeys to get even closer.

On the more pricey side, Switzerland has world-famous panorama train journeys. All of these countries also have cable cars bringing you to very high mountains with minimal effort (Aiguille du midi, Skyway Mont-Blanc, Wildspitz Bahn, Klein Matterhorn, Zugspitze…) There is also an elevator at the Jungfrau and cogwheel trains in Saint-Gervais, Chamonix or Zermatt (Gornergrat).

Finally, if you go to one of these places, also consider cable cars on the other side of the valley or to lower moutains in the area, they can offer great views of the highest peaks (I am thinking of something like the Brévent in Chamonix but you will need to narrow it down to get specific advice).

Some mountain ranges are impressive from a distance.
The Alps are from some areas but most of the time they are surrounded by lower mountains and hills.

So your best 'view' of the Alps is from 'inside'.
You can often get train tickets, timed to one train only, which have a good price for international travel.

In Italy, Switzerland and Austria, the main regions of the Alps, you will find youth hostels which, while still not cheap, are less expensive than hotels and often do offer the option to make your own meals, which is also cheaper than going out for dinners in a restaurant.

In your case, no car and a relatively low budget, I would select one city or even village in the Alps and work out a train journey to there and from there to an other town (inside or outside the Alps, as your budget allows you) and by booking early you can see what it will cost you if you buy your tickets on the day you check it.
The earlier you book, the better chance to get cheap tickets. For hostel (and hostel) prices you can look on booking sites, there are several. Search for 'hostel' or 'hotel' and the name of the town or village and most likely a booking site as well as direct links to hostel will be in the results.