Edinburgh’s Summer of Discounts : Students in Scottish capital this season can find savings on lodging, walking tours, bus travel and museum admissions.

Historic Edinburgh presents budget travelers with a rucksack full of summer bargains. Many museums are offering free admission, and there are inexpensive walking tours, hostel-style accommodations and a low-cost bus service designed to transport backpackers to other areas of Scotland.

Edinburgh is easy to reach from London. Seventeen trains cover this route daily, and the high-speed InterCity 225 service--which can be used by BritRail Pass holders--makes the 393-mile trip in less than four hours.

A good place to start your visit is the Edinburgh Tourist Information Centre at 3 Princes St., just outside the central railway station. Here you can get a free brochure on independent hostels throughout Scotland, details on what to see and do around Edinburgh, comprehensive city and countryside maps, and information on walking tours.

Companies such as Robin’s Tours offer entertaining themed walking tours and give student discounts. For example, on the Dr. Jekyll tour, conducted at 9 p.m. nightly until October through the dark streets of Edinburgh’s old town, students can get about 75 cents off the 3 rate (making the price about $4.25 U.S.).

Hostels offering student-style budget accommodations include:

* Bruntsfield Hostel, 7 Bruntsfield Crescent, from U.S. phones 011-44-31-447-2994. The Bruntsfield has 170 dormitory beds and is affiliated with Hostelling International. Travelers over 18 pay about $11 per night.

* Eglinton Hostel, 18 Eglinton Crescent, tel. 011-44-31-337-1120. It has 184 dormitory beds and is also affiliated with Hostelling International. Travelers over 18 pay about $14.50 per night.

* Belford Youth Hostel, 6/8 Douglas Gardens, tel. 011-44-31- 225-6209. It has 40 beds each for men and women and some family rooms. Dormitory beds cost about $11 and up per night. Beds in double rooms start at $19 per night.

The most popular do-it-yourself walking tour begins at Edinburgh Castle, which still houses the royal jewels of Scotland (admission about $5; no student discount). Downhill from the castle, along what is known as the Royal Mile, you’ll pass Lady Stair’s House--a small museum (free) devoted to literary native sons Robert Burns, Sir Walter Scott and Robert Louis Stevenson.

Further along, at 163 Canongate, you’ll find the People’s Story Museum (also free). It tells of the life and work of ordinary people in Edinburgh since the late 18th Century. Eventually, you will wind up at the Palace of Holyroodhouse, the Queen’s residence in Scotland.

Other free museums and galleries in Edinburgh include:

* The Royal Museum of Scotland, Queen Street. The museum’s collection dates to the Stone Age.

* The Royal Museum of Scotland, Chambers Street. The collection includes primitive art from around the world.

* Scottish National Portrait Gallery, 1 Queen St. This gallery features portraits of major figures in Scottish history.

For travelers who want to visit the Scottish countryside but have not purchased transportation, or whose rail pass is about to run out, here are some options:

* Go Blue Banana, a minibus service, covers a circular route from Edinburgh to the Isle of Skye and back. You can buy a ticket to cover the whole circuit for about $70 and take as long as you want to complete your journey. The service is operated by backpackers for backpackers. Stops are made at interesting sights along the 560-mile route, and the buses pick up and drop off at about 30 hostels along the way. For more information, contact Go Blue Banana, 12 Rutland Square, Edinburgh, tel. 011-44-31-228-2281.

At the Scottish Youth Hostel Assn. district office, 161 Warrender Park Road, tel. 011-44-31-229- 8660, travelers can buy packages that combine rail and bus travel with youth hostel accommodations.

The Scottish Wayfarer package, for instance, includes a pass for eight or 15 days of unlimited second-class rail travel on most Caledonian MacBrayne’s West Coast ferries, one-third off travel on participating bus services, and sea passage to the Orkney Islands. Price for the eight-day package is about $245, 15 days about $375.