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LingakuLingaku[subject] + 进行 + [verb]
对 + [noun] + 进行 + [verb]
[subject] + 开展 + [noun]
[subject] + 展开 + [verb] / [noun]
进行, 开展 and 展开 all take an activity as their object, and all three belong to written Chinese: reports, news, official notices. 进行 carries out a single piece of work, 开展 runs a programme over a period, and 展开 has something open out and get going in full.
进行 adds almost no meaning of its own. It takes a two-syllable verb as its object – 调查, 研究, 检查, 讨论, 比赛 – and makes the sentence sound formal. The thing being discussed or examined can't follow as a second object; it moves in front with 对.
A one-syllable verb or a plain noun can't follow 进行 at all.
开展 starts something and keeps it growing. Its subject is usually an organisation, and its objects are things that run for weeks or years: 活动, 工作, 合作, 运动.
A single meeting or test is too small for 开展.
展开 first means to unfold something folded: a map, a letter, wings. Used about an activity, it says the activity started and spread – a debate, a search, a competition. It can stand without an object, with the activity as subject.
进行 describes the activity as it happens; 展开 marks its start and its scale.